Mathematics practice. mathematics curreculum pptx

Thinking about Thinking & Learning
As you learned in the Orientation module, students engaged in metacognition
reflect on
● What they already know about a topic
● What they need to learn
● What strategies or actions will help them learn
● How they will apply, communicate, and retrieve the results of their learning
Several of the Standards of Mathematical Practice (SMP)
specifically address components of student metacognition.
1

The Standards of Mathematical Practice: WHY
Real-life Mathematics Application
The eight SMP are the antidote to teaching mathematics as a series of "plug and
chug" procedures ("Do this for this kind of problem"). According to the Association
for Supervision and Curriculum Development (ASCD):
Most real-life mathematics does not come in worksheet form. Those who make
sense of real-life problems start by trying to figure out what is meant in the
given situation. What is being asked? What information is given, what needs to
be found, and which mathematical procedures and tools will lead to that
information? Our students will approach word problems with questions like
these once they master the mathematical practices. They will not only be able
to solve mathematically rich problems, but they will also appreciate math's
usefulness. 2

The Standards of Mathematical Practice: WHY
Several math practices focus on doing math, talking about math and on
strategically thinking about math. Well-equipped students use mental habits
that lead them toward:
● describing problems (and solutions) in precise ways.
● subdividing and exploring problems by posing new and related problems.
● “playing” concretely (or with thought experiments) to gain experience and
insights from which some regularity or structure might be derived.
● seeking and articulating underlying structure that might relate new
problems to ones that have already been solved.
● choosing approaches both strategically and flexibly. 3

The Eight Standards of Mathematical Practice (SMP)
SMP1: Make sense of problems and persevere in solving them.
SMP2: Reasoning abstractly and quantitatively.
SMP3: Construct viable arguments and critique the reasoning of others.
SMP4: Model with mathematics.
SMP5: Use appropriate tools strategically.
SMP6: Attend to precision.
SMP7: Look for and make use of structure.
SMP8: Look for and express regularity in repeated reasoning.
4
The Eight
Standard
s of
Practice

SMP1: Make sense of problems and persevere in
solving them.
Students proficient in this practice will be able to
● start by explaining to themselves the meaning of a problem and looking for
entry points to its solution.
● analyze givens, constraints, relationships, and goals.
● make conjectures about the form and meaning of the solution and plan a
solution pathway rather than simply jumping into a solution attempt.
● consider analogous problems, and try special cases and simpler forms of the
original problem in order to gain insight into its solution.
5

solving them.
● monitor and evaluate their progress and change course if necessary.
● perform algebraic transformations and/or change the viewing window on their
graphing calculator to get the information they need depending on the context
of the problem.
● search for patterns or trends and use those to explain relationships among
equations, verbal descriptions, tables, graphs and diagrams.
6

solving them.
● use concrete objects or pictures to help conceptualize and solve a problem
(especially important for less experienced students).
● check their answers to problems using a different method and continually ask
themselves, “Does this make sense?”.
● understand the approaches of others to solve complex problems and identify
relationships between different approaches.
7

Five Different Ways to Represent a Function
8
With a graph
With words
With a diagram
In a table
With an
equation

Actions That Encourage Students to Make Sense of
Problems and Persevere in Solving Them (SMP1)
What instructors can do:
● Allow students time to initiate a plan. Use question prompts as needed to
assist students in developing a pathway.
● Question students to see if they’ve made connections to previous solution
attempts and/or tasks to make sense of the current problem.
● Ask students to defend and justify their solutions by comparing solution paths.
● Differentiate content and instruction to keep all levels of students challenged
during work time.
9

SMP2: Reason abstractly and quantitatively.
● Mathematically proficient students make sense of quantities and their
relationships in problem solving situations.
● Mathematically proficient students bring two complementary abilities to bear
on problems involving quantitative relationships:
1. The ability to decontextualize - Move from specific, often real-life cases, to
more abstract representations using, for example, equations, inequalities, graphs,
or tables.
2. The ability to contextualize - Move from an abstract to a specific situation by
pausing as needed during the manipulation process to consider what a given
model means or how that model can be applied to real-life situations.
10

A Low-Level Contextualized Example
11
Dee collects drink bottles then returns them for recycling. She receives 10
cents for each bottle. How much money would she get if she recycles 30
bottles?
A student could calculate this answer by adding 10 thirty times (very concrete) or by
multiplying 30 x 10 (a little less concrete). Alternatively, dimes (or other counters)
could be used to model this situation.

The Previous Example: Shifting to Decontextualizing
Looking at the equation and graph above, students who can decontextualize
understand the relationship between x and y without a context or specific
quantities. It does not matter exactly what x or y represents.
12
y = 0.1x

Decontextualizing and Contextualizing:
Intertwined Processes
After students read the problem and understand the context of the quantities,
the next steps might be something like the following:
1. Translate the words into math - numbers, symbols, or operators
(contextualize).
2. Write an expression, equation, inequality, function, or system to show
the relationship between the parts (decontextualize).
(Continued on next slide)
13

Decontextualizing and Contextualizing:
Intertwined Processes
3. Crank out the solution(s) - compute, evaluate, solve equation(s) and
systems, simplify expressions, etc. (context to general and back to context).
4. Go back to the original problem to interpret and understand what the
solution(s) mean(s) (contextualize and decontextualize).
5. Extend the solution approach to other similar problems to generalize the
approach for problems that might be related (contextualize and
decontextualize).
14

SMP2: Reason abstractly and quantitatively.
Students proficient in quantitative reasoning are able to create a coherent
representation of the problem at hand by:
● Considering the units involved.
● Attending to the meaning of quantities, not just how to compute them.
● Flexibly using different properties and number operations.
15

SMP3: Construct viable arguments and critique the
reasoning of others.
Students proficient in SMP3 are able to:
● Construct mathematical arguments based on stated assumptions, definitions,
and previously established results.
● Make conjectures and build a logical progression of statements to explore the
validity of their conjectures.
● Analyze situations by breaking them into cases.
● Recognize and use counterexamples.
● Justify their conclusions and communicate their results to others.
16

SMP3: Construct viable arguments
and critique the reasoning of others.
In addition, students proficient in SMP3 are able to:
● Compare the effectiveness of two plausible arguments, distinguish correct
logic or reasoning from that which is flawed, and, if there is a flaw in an
argument, explain what it is.
● Listen or read the arguments of others, decide whether they make sense,
and ask useful questions to clarify or improve the arguments.
17

SMP3: Construct viable arguments and critique the
reasoning of others.
It has been said among educators that students retain:
● 5% of what they hear in a lecture.
● 50% of what they are allowed to practice (as in homework).
● 90% of what they see and do.
Instruction based on seeing and doing, student interaction, and teamwork provides
many opportunities to engage with SMP3. SMP3 addresses learning as a social
process and acknowledges that learners have much to gain by reflecting on not
only their own thought processes, but also those of other learners.
18

The modeling process allows an approach that is investigative rather than
prescriptive and grants the instructor the freedom to act as a “guide on the side”
rather than a “sage on the stage” as students and instructor explore the question
together.
The result of the above approach is a dialogue-based and learner-centered
process of discovering what math best represents a particular question or
problem.
19

Students proficient in SMP4 are able to
● Apply math to everyday life, the community, and
the workplace.
● Make assumptions and approximations to simplify a situation and
understand that those assumptions may need to be modified later.
● Identify important quantities in a practical situation and map their
relationships using tools such as graphs, charts, equations, formulas, and
tables.
20

It is possible to modify a ready-made math problem into a modeling math problem
by adding labels, context, and an interpretive or application aspect to it. However,
ideally, modeling with mathematics involves developing the math after the question
is posed, not beforehand.
A second important aspect of modeling with SMP4 lies in giving students
opportunities to both interpret the problem and have choices and input in the
process of determining the solution.
21

The Steps of Modeling With Mathematics (SMP4)
22
1. Specify a problem: Identify something in the real world that we wish to know,
do, or understand.
2. Make assumptions and identify variables: Decide which components of the
real-world problem are important enough to distill into mathematical variables.
3. Obtain a mathematical formulation: Translate the variables and their
relationships to one another into mathematical terms; e.g., a model.
4. Compute: Use the model to see what results and insights are gained.
(Continued on next slide)

The Steps of Modeling With Mathematics (SMP4)
(Continued)
5. Assess the solution: Analyze whether the solution addresses the problem
or not. Does it make sense when translated back into the real world? Do we
discover that we left important variables out of the model?
6. Iterate: Repeat the model-building process to refine and/or extend the model.
7. Implement the model: Report the results to others and implement the
solution.
23

Modeling With Mathematics: An Example Problem
Gas prices often seem to change on a daily basis. If you know locations
and prices for all nearby stations, from which station should you buy gas?
Under what conditions would you be willing to drive to the more distant gas
station? Create a mathematical model that displays under what conditions
it is worth it to drive to the cheaper gas.
Note that the question is broad, non-specific, and open to interpretation. This is
done on purpose so that students can brainstorm what assumptions, and therefore
what variables, are most important to consider when developing an associated
mathematical model.
24

An Example of a Student-Derived Model for the Gas
Station Problem Using Unit Analysis (SMP4)
Let’s say that a group of students tackling the gas station problem identified the
main considerations as
● The distances between the stations with
various price points
● The fuel economy of the car they are driving
● The price of gas at various stations
25

The students then contextualized the problem by adding some specific
information:
● Station A is on the regular route between home and school, thereby requiring
no extra travel but with a price per gallon of $3.80.
● Station B is 8 miles away but gas costs only $3.65 per gallon.
● Ten gallons of gas will be purchased.
● The fuel economy of their car is 30 miles per gallon.
26

Finally, they decided on a model using unit analysis and did these computations:
Station 1: $3.80 per gallon - $3.65 per gallon = $0.15 difference per gallon between
the stations
$0.15 x 10 gallons = $1.50 more to buy gas at Station 1 vs. Station 2
Station 2: 8 miles x 1 gallon x $3.65 = $0.97 is the cost to drive to Station 2
30 miles 1 gallon
Solution: $1.50 - $0.97 = $0.53 saved by driving to Station 2
27

Students who use SMP5 successfully are able to
● Make sound decisions about the use of specific tools (Examples might
include: calculator, concrete models, digital technologies, pencil/paper,
ruler, compass, protractor).
● Use technological tools to visualize the results of assumptions, explore
consequences, and compare predictions with data.
● Identify relevant external math resources (digital content on a website) and
use them to pose or solve problems.
● Use technological tools to explore and
deepen understanding of concepts.
28

Mathematics tools fall into three general categories: physical, digital, and
cognitive. Allowing students a choice of tools leads to greater independence.
Examples of physical tools: counters, markers or crayons, whiteboards,
toothpicks, pattern blocks, Multilink cubes, dried spaghetti noodles, cash register
tape, masking tape, algebra tiles, pan balances, rulers, cheese crackers,
transparencies.
29

Examples of digital tools: Desmos online graphing calculator, Math Playground,
spreadsheets (e.g., Excel, LibreOffice), YouTube, Poll Everywhere, Geogebra,
PhET Interactive Simulations. There are literally thousands of digital tools
available, so this list certainly is not exhaustive.
Examples of cognitive tools: graphic organizers, communications software such
as teleconferencing programs, online collaborative knowledge construction
environments, multimedia/hypermedia construction software, lists, mnemonic
devices, models.
30

Attention to precision is an overarching way of thinking mathematically and is
essential to teaching, learning, and communicating in all areas of mathematical
content across all levels.
Part of this standard relates to students calculating accurately and efficiently.
Students who calculate accurately and efficiently:
● Have command over simple arithmetic operations (especially those with
benchmark numbers).
● Know when to employ a tool such as a calculator or spreadsheet.
● Use estimation to check for the reasonableness of computation.
31

Mathematically proficient students communicate precisely to others.
● They use clear definitions in discussion with others and in their own
reasoning.
● They state the meaning of the symbols they choose, including using
the equal sign consistently and appropriately.
● They can defend the definitions in their work and identify flaws in
thinking that need to be modified, refined, and corrected.
32

Mathematically proficient students are careful about specifying units of measure
and labeling axes to clarify the correspondence with quantities in a problem.
Here’s a situation to think about: Michael Phelps’ record time for swimming 100
meters is 47.51. Without units, that time makes no sense. Perhaps it took Michael
47.51 hours to swim that far - that is extremely slow and certainly not an Olympic
gold medal winning time!
33

Students who attend to precision
● Use correct and specific mathematics vocabulary.
○ Understand the differences between linear, square, and cubic units and
use the correct terminology for measurements of length, area and
volume.
● Use symbols correctly.
○ Know why it is incorrect to write 13 + 7 = 21 + 5 = 26 x 2 = 52.
● Attend to the degree of precision needed in the context.
○ “My calculator says 3.581279, but since I’m asked to find the number of
centimeters, that’s not a number that makes sense for this problem. I’ll
say 3.6 cm.”
34

Mathematically proficient students
● Look closely for the overall structure and pattern in mathematics.
● Apply general mathematical rules to specific situations.
● See complicated things as single objects or as being composed of several
objects.
35

● Example - putting off calculation until one sees the overall structure-helps students
notice that they don’t have to find the common denominators for
1¾ – ⅓ + 3 + ¼ – ⅔
but can simply rearrange the terms to make it easy to compute mentally.
● Example - As students engage in algebraic thinking, they can see structure. In the
equation below, they can see that something plus 2 equals 20 and conclude by
using common sense and not just “rules” that 3(5x-4) equals 18.
36

SMP8: Look for and express regularity in repeated
reasoning.
Mathematically proficient students are able to
● Notice if calculations are repeated; look both for general methods and for
shortcuts.
● Maintain oversight of the process as they work to solve a problem, while
attending to the details.
● Continually evaluate the reasonableness of their intermediate results.
37

● Establish clear mathematics goals to focus learning and attend to logical
progressions.
● Implement tasks that promote reasoning, problem solving and discussion, and
allow multiple entry points and varied solution strategies.
● Engage students in making connections among mathematical representations to
deepen understanding of mathematics concepts and procedures
as tools for problem solving.
Using the SMP to Enhance Mathematics Learning:
Some Effective Teaching Strategies
38

● Facilitate meaningful mathematical discourse to build shared understanding of
mathematical ideas by analyzing and comparing student approaches and
arguments.
● Pose purposeful questions to assess and advance students’ reasoning and
sense making about important mathematical ideas and relationships.
● Build procedural fluency from conceptual understanding so that students, over
time, become skillful in using procedures flexibly as they solve
contextual and mathematical problems.
39

● Support productive struggle by providing students with opportunities to
grapple with mathematical ideas and relationships.
● Elicit and use evidence of student thinking as a formative assessment to
evaluate progress toward mathematical understanding and to adjust
instruction continually in ways that support and extend learning.
40

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