The document discusses mathematical skills and higher order thinking skills (HOTS) in mathematics. It defines arithmetic skills such as addition, subtraction, multiplication and division. It also discusses geometric skills and interpreting graphs and charts. The document then defines HOTS as including skills such as problem solving, reasoning, communication and conceptualizing. It provides examples of each skill and discusses the importance of incorporating HOTS into mathematics teaching to better prepare students. The document concludes by providing suggestions for how to improve students' HOTS through revising textbooks and using open-ended testing.

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MATHEMATICAL SKILLS
I) ARITHMETIC SKILLS
Arithmetic
 “Mathematics is the queen of sciences
and Arithmetic is the queen of
Mathematics“ .
 Greek word
 The science of numbers and art of
computing
 Oldest branch of the subject
Early Math Skills

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Before starting school, most children develop an
understanding of addition and subtraction through
everyday interactions.
Key Math Skills for School
• Number Sense (forward and backward)
• Representation (mathematical ideas using words,
pictures, symbols, and objects )
• Spatial sense (Geometry, ideas of shape, size, space,
position, direction and movement.)
• Measurement (finding the length, height, and
weight of an object )
• Estimation (more, less, bigger, smaller, more than,
less than)
• Patterns (learn to make predictions, to understand
what comes next, to make logical connections )
• Problem-solving (using past knowledge and logical
thinking skills to find an answer )

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Top 4 Basic Math Skills Students Should Learn
• Problem solving (to develop analytical thinking)
• Applied Math (applying math in everyday
situations)
• Estimation and approximation (to use almost every
day)
• Necessary computational skills
Fluency
• Math fluency is defined as the “ability to recall basic
math facts.”
• By enhancing fluency, one can develop more
confidence and less dependence.
Tips for speed and accuracy
• Do speed addition and subtraction tests
(50 problems in 3 minutes)
• Memorize multiplicationtables up through 15
(extend to 15)

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• Recall your multiplicationtables for division
• Simplify calculations into smaller,easier
calculations
Fundamental Arithmetic Ideas
• Addition
• Subtraction
• Multiplication
• Division
• Fractions
 Latin word
 ‘break’
• Decimal Fractions
 Simon Stevin,Belgium,1584
 Decimal point by Napier,1617
• Ratio
• Proportion

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• Percentage
• Simple interest
• Compound interest
• Area
• Square /Rectangle
• Parallelogram
• Triangle
• Cross roads
• Circular roads
• Circle
• Volume
• Cylinder

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II) GeometricSkills
Geometry
 Greek word
 ‘earth measurement’
 The science of lines and figures (or) The
science of space and extent
 Deals with the position ,shape and size of
bodies.
 It is the picturized arithmetic / algebra
3 stages of teaching Geometry
• Practical Stage
 common geometrical concept
• Stage of Reasoning
 to prove theorems and exercises
• Systematic Stage
 acquisition of mastery in reasoning

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Traits of mathematically able children
Ability to make and use generalizations often quite
quickly.
 Rapid and sound memorization of mathematical
material.
 Ability to concentrate on mathematics for long
periods.
 An instinctive tendency to approach a problem in
different ways
 Ability to detect unstated assumptions in a problem.

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III) Interpretationof graphs and charts
Graphs
 Graphs are especially useful for presenting
quantitative data.
 A graph is a visual form of data from a table.
 Graphs are ideal for communicating scientific
information.
 A graph can make it easier to analyze and interpret
the information you have collected.
Features of a graph
• title
• grid
• horizontal axis or X-axis
• vertical axis or Y-axis

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Differentkinds of graphs
1)Line graphs
 To show how one variable affects another.
 Line graphs are useful in that they show the
relationship between two variables.
2)Bar and column graph
 Both used to show categories of data that has
been counted.
 In a column graph, the height of the column
shows the number of individuals.
 In a bar graph, the length of the horizontal bar
represents the number of individuals.
3)Histograms
1) Histograms look similar to column graphs
but are different because each column
represents a group of related data.
2) The height of the column shows the
number of individuals counted, like a
column graph.

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4) Pie graphs
1) Pie graphs are useful for showing the
percentage composition of various
categories that are unaffected by each
other.
2) Computer programs can present pie
graphs in many different, interesting shapes
5)Scattergrams
1) Scattergrams are graphs that are used to
find patterns in some kinds of data.
2) The information about each individual or
test is plotted as a separate point.
Choosing a graph type
Each time you need to draw a graph, think about the
different kinds of graphs. Decide which kind of graph will
make your data the easiest to analyses and interpret.
Sometimes there are several kinds of graphs
that would be suitable.

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For example,
Usually you can use a bar graph, a column graph or a
pie graph to present the same kinds of information. A
line graph and a histogram can show the same data.

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IV) H O T Skills
‘H O T Skills ‘
- Higher Order Thinking Skills
History
Human thinking skills can be classified into two
major groups:
Lower order thinking skills (LOTS)
Higher order thinking skills (HOTS).
 LOTS have three aspects
Remembering
Understanding
Applying. (The first 3 of Bloom’s taxonomy)
HOTS have three aspects
Analyzing
Evaluating
Creating. (The last 3 of Bloom’s taxonomy )

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Why teach Higher Order Thinking Skills(HOTS)?
• “Higher-order thinking skills are valued because they
are believed to better prepare students for the
challenges of adult work and daily life and advanced
academic work. “
- Pogrow
• Higher-order thinking may also help raise
standardized test scores.
• The revised secondary mathematics curriculum has
shifted its emphasis to the fostering of HOTS.
Higher Order Thinking Skills
• Higher-order thinking is thinking on a higher level
than memorizing facts or telling something back to
someone exactly the way that it was told to you.
• HOTS is the highest part in Bloom’s taxonomy of
cognitive domain.

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Five fundamental HOTS
1.Problem solving skills
2. Inquiring skills
3. Reasoning skills
4. Communicating skills
5. Conceptualizing skills.
1.Problem solving skills
• According to National Council of Teachers of
Mathematics (NCTM),
“ problem solving is a process of applying
previously acquired knowledge to new and
unfamiliar (or unforeseen) situations.”
• Four phases are ,
- understanding the problem
- devising a plan of solving the problem
- carrying out the plan
- examining the reasonableness of the
result and making evaluation.

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2. Inquiring skills
• Inquiring involves discovering or constructing
knowledge through questioning or testing a
hypothesis.
• The essential elements are,
Observation
analysis
summarizing
verification
3. Communicating Skills
• Communication involves receiving and sharing ideas
and can be expressed in the forms of numbers,
symbols, diagrams, graphs, charts, models and
simulations.
• 2 types :
oral and written
• 3 stages :
construct
refine

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consolidate
4. Reasoning Skills
• Reasoning is drawing conclusions from evidence,
grounds or assumptions.
• 2 types :
inductive reasoning
deductive reasoning
5. Conceptualizing Skills
• Conceptualizing involves organizing and reorganizing
of knowledge through perceiving and thinking about
particular experiences in order to abstract patterns
and ideas and generalize from the particular
experiences.
 The formation of concepts involves classifying and
abstracting of previous experiences
Three components in HOTS
(1) critical thinking skills
(2) creative thinking skills
(3) systems thinking skills.

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Nine factors that comprise HOTS
• the use of mathematical concepts
• the use of mathematical principles
• impact predicting
• problem solving
• decision-making
• working in the limits of competence
• trying new things
• divergent thinking
• imaginative thinking.
Uses of Higher-order thinking
• to manipulate information and ideas
• to synthesis, generalize, explain, hypothesis (or)
arrive at some conclusion or interpretation.
• to solve problems and discover new meanings and
understandings.
• i.e., the teacher is not certain what will be produced
by students.

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Note the following
1)There is no simple, clear and universally accepted
definition of HOTS.
2) The five HOTS cannot be easily isolated from each
other in mathematicalwork.
3) HOTS can be taught in isolation from specific
contents, but incorporating them into content areas
seems to be a popular way of teaching these skills.
4) Computer provides an excellent tool for teaching
HOTS
Questions and investigations
• Closed question - 6 + 4 = ?
• Open question - What numbers could you add
together to make 10?
• Investigation - There are chickens and calves
in the farmers paddock. If there are 24 legs
altogether, how many of each animal could there
be?

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HOT and investigation
Example :
Noah counted 24 legs as the animals walked
into the ark.
What types of animals might they have been?

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Levels of thinking
Learning of rules and facts
Applying rules and formulae in standard situations
Solving specific problems in novel situations
Investigating issues and problem situations
Relationshipbetween HOTS and student performance
in Mathematics
• There is a linear, positive and strong relationship
between HOTS and the GPA of students.
• Students with high level of HOTS are expected to
succeed in their next study in study program of
mathematics education.
• Students who have high HOTS tend to get high GPA
in mathematics instruction.
• Therefore, the value of HOTS can be used as an
indicator in the selection of new students.

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• In order to thrive in learning mathematics, students
should have high level of HOTS.
To improve student’s HOTS
• To improve student’s HOTS , revise textbooks used in
mathematics learning in primary and secondary
schools.
• The mathematics textbooks used in Indonesia should
promote student’s critical and creative thinking.
• Examples and practice tests provided should be able
to train students to think critically and creatively by
using open-ended test.
• The open-ended test is a test used as an instrument
in this study to measure students' HOTS

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