what is mathematics and what is being taught?achieve, jason zimba, 2012, curtis center. what content...

What is mathematicsand what is being taught?

La Canada February 25, 2016

Heather Dallas

Crime is No Match for Math} Algorithms developed the UCLA Mathematics

Department more accurately predict crime hotspots than trained analysts. In Los Angeles, the algorithm program predicted the location crimes 2.2 times more accurately than the analysts’ predictions. The Los Angeles Police Department has adopted the algorithm-based program for 19 of its 21 divisions that helps deploy officers most effectively.

http://admin.magazine.ucla.edu/depts/quicktakes/crime-is-no-match-for-math/

UCLA mathematicians bring snow to life for Disney's 'Frozen'} Professor Teran found that a good model to create

snow in computer graphics didn’t exist. “Snow reacts differently than other materials…If you squish snow, it’s going to get harder. But if you stretch it, it gets weaker and breaks apart. We took all that into consideration for our model.”

http://newsroom.ucla.edu/stories/math-wizards-create-snow-for-disney-263913/

What is Mathematics?

AxiomsDefinitionsTheorems

AlgorithmsProblems

& Solutions

Mathematical Habits of Mind:

Problem SolvingRecognizing structureCreating definitions

ProvingQuestioningRepresenting

The Common Core “Standards of Mathematical Practice”

Is problem solving (MP 1) being taught?

40% of instructional time in the Japanese TIMMS video study classrooms was spent on solving

problems compared to 0.7% in U.S. classrooms.

100% of that 0.7% , U.S. teachers reduced the problem into an exercise.

(Stigler, 1995)

TIMSS 1995

Achieve, Jason Zimba, 2012, Curtis Center

12th out of 26

TIMSS 1995


28th out of 42

IF JUSTIFICATION IS ABSENT FROM OUR CLASSES,

WE ARE NOT TEACHING MATHEMATICS.

Hung Hsi Wu, Berkeley Mathematics Professor, Mathematical Sciences Research Institute

Critical Issues in Mathematics Education Workshop, 2011

Is proof (MP 3) being taught?

53% of the Japanese lessons used proof-based reasoning, whereas the comparable statistic for the U.S. lessons stood at zero.

J. W. Stigler et al., “TIMSS Videotape Classroom Study.”

Is proof (MP 3) being taught?

“There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from. The student who can explain the rule understands the mathematics.”

Common Core State Standards for Mathematics,Introduction, p. 3

Math Practice 3Construct Viable Arguments

Is applied mathematics (MP 4) being taught?

NCTM Focus in High School Mathematics:Reasoning and Sense Making, p.70

Common Core Content Standard 8.G.7

Apply the Pythagorean Theorem to determine unknown side

lengths in right triangles in real-world and mathematical

problems in two and three dimensions.

Math Practice 4Model with Mathematics

Smarter Balanced Assessment ConsortiumTask 43052

Math Practice 4Model with Mathematics

What Content is Being Taught?

AxiomsDefinitionsTheorems

AlgorithmsProblems

& Solutions

Mathematical Habits of Mind:

Problem SolvingRecognizing structureCreating definitions

ProvingQuestioningRepresenting

Schmidt, Houang, & Cogan, “A Coherent Curriculum: The Case of Mathematics.” (2002).

Topics intended at each grade by at least

2/3 of A+ countries

Topics intended at each grade by at least

2/3 of 21 U.S. states


} “On average, the U.S. curriculum omits only 17 percent of the TIMSS grade 4 topics compared with an average omission rate of 40 percent for the 11 comparison countries.

} The United States covers all but 2 percent of the TIMSS topics through grade 8 compared with a 25 percent noncoverage rate in the other countries.

Ginsburg et al., 2005



} High-scoring Hong Kong’s curriculum omits 48 percent of the TIMSS items through grade 4, and 18 percent through grade 8.

} Less topic coverage can be associated with higher scores on those topics covered because students have more time to master the content that is taught.”

Ginsburg et al., 2005


Common Core: Internationally Benchmarked

} FOCUSEDFewer topics per grade level

} COHERENTTopics within a grade link together and

topics build logically over the grades

} RIGOROUSIn major topics, students develop:

} Conceptual understanding} Procedural skill and fluency

} Applications



The coherence and sequential nature of mathematics dictate the foundational skills that are necessary for the learning of algebra. The most important foundational skill not presently developed appears to be proficiency with fractions (including decimals, percents, and negative fractions). The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in algebra can be expected.”

Final Report of the National Mathematics Advisory Panel (2008, p. 18)

Coherence: Connections across & within grades

} K.G.6 Compose simple shapes to form larger shapes. For example, "Can you join these two triangles with full sides touching to make a rectangle?”

} 1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.


} 2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

} 2.MD.6 Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.


} 3. G. 2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape

Develop understanding of fractions as numbers.} 3. NF. 1 Understand a fraction 1/b as the quantity

formed by 1 part when a whole is partitioned into bequal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

} 3. NF. 2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.


RIGOR: What does it mean to be good at math?

} Fluency (speed & accuracy)

} Conceptual Understanding (why what I’m doing works)

} Applications

6.EE.9Use variables to represent two quantities in a real-world

problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity,

thought of as the independent variable.

Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving

motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d=65t to represent the relationship between distance and time.

Rigor: Understanding, Application & Fluency

8.F.1Understand that a function is a rule

that assigns to each input exactly one output.

8.F.4Construct a function to model a linear relationship

between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading

these from a table or from a graph. Interpret the rate of change and the initial value in terms of the situation it

models.


8.EE.6Use similar triangles to explain why the slope is the

same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx

for a line through the origin and y = mx+b for a line intercepting the vertical axis at b.


F.IF.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of

the range.

If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the

input x.

The graph of f is the graph of the equation y = f(x).

Rigor: Understanding, Application, & Fluency

Overview of LC Math Pathways 2015-16

Progress Report –LC Math 1 Courses

LC Math 1 Adv.Dr. Carruthers and Mr. McDermott

5 sections offered

Avg. 28 students per class

PLC and common preparatory period

LC Math 1Ms. DiFiore and Mr. Szamosfalvi

5 sections offered

Avg. 25 students per class

PLC and common preparatory period

New Courses Being Developed

�New Math Teams Formed

�New Math Consultant: Heather Dallas - Executive Director - UCLA Curtis Center

�Math Compaction Being Realized

�4 Productive Planning Meetings�Concepts Within Each Compaction �Standards Alignment

New Classes Being Developed - Multiple Drafts

LCHS and LCFEF Collaboration

�Early Planning Meetings

�Course Offerings in Summer 2016�LC Math 1,LC Math 1 Bridge

Course, Geometry, Algebra II/Trig

�Sharing of LC Math 1 Curriculum Outline and Materials

�Development of LC Math 1 Bridge Course

LC Math 1 Bridge Class

If Parents Are Considering Summer School…

●Math in Summer School –Intensive Study

●Options available to students

●High School Credits earned through Summer School

Next Steps

● Submit the new courses to UC for approval

● Start registration

● Select teachers to teach the courses

●Work with the consultant to design the pacing guides and assessments for these courses

Next Steps

• Align Science course descriptions to reflect changes in math sequencing

• Continue to communicate with students and their families on math pathways

A Joint Position Statement of the MAA and the NCTM on Teaching Calculus (2012)

…the ultimate goal of the K–12 mathematics curriculum should not be to get students into and

through a course in calculus by twelfth grade but to have established the mathematical foundation that will enable students to pursue whatever course

of study interests them when they get to college.

http://www.maa.org/news 2012_maanctm.html

39

From Alhambra USD to UC Board of Admissions and Relations with Schools Chair, Prof. Jacobs

…Our parents feel, not having two year AP Calc (AB and BC), we are depriving students' opportunity to take AP exam in 11th grade, thus, reducing their compatibility in college application. Would you share your taking on this? Our goal is to serve our students in the best way we know but we will not jeopardize their future if parent's concern has its ground.

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From UC Board of Admissions and Relations with Schools Chair, Prof. Jacobs to Alhambra USD

41

Your plan sounds absolutely right on target to me. The districts I have worked with have found that accelerating students into Algebra 1 in 7th grade is a serious mistake because a good many students don’t stick with math past ninth or tenth grade because they get burned out or struggled more than anticipated. Then they are not as well prepared for the University as they could have been. And while some students do two years of calculus in grade 11 and 12 the risk and the damage to the others is too high to justify it in my opinion.


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It is very difficult to predict who will be successful four years down the road when a child finishes grade 6 as too many changes go on in the intervening year. Personally, with the CCSS offering a well-designed 8th grade course I wouldn’t push students except the very strongest ahead of that course either. We want as many students to take 4 years of math in high school and if they get accelerated a good number stop too soon. CSU agrees with this point of view as well (they emphasize this in our yearly joint admission meeting), as they have found students who stop math after grade 10 have a very high likelihood of ending up in their remedial course.


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As far as taking AP tests, it is more important that students take an all around challenging curriculum than race to and extra math AP course. In admission to UC the readers are aware of AP courses, but not so much the test scores, and so if the student had honors courses that lead to an AP course in grade 12 that is going to be about as strong as they can do. If this rush leads to weaker grades overall the student is much worse off. What I’ve seen over the past 25 years teaching at UCSB is that the BC Calculus really doesn’t move students much further forward mathematically (perhaps one quarter, ten weeks, at most)…


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and I would much rather see them have a thorough preparation including applications of geometry in the years prior to calculus. The students who don’t have these preliminary experiences have tremendous difficulty in my differential equations and multivariable calculus courses (certainly those in my classes when I interview them). I know it is hard to persuade parents that their students shouldn’t race to get calculus, but I really wish they wouldn’t. So I like your plan. Keep up the good work.

To Accelerate, or NotProf. Hung Hsi Wu, UC Berkeley Mathematics

} At present, the school math curriculum [does not] lead to the kind of mathematics learning that will get our nation out of its present educational doldrums.To improve, we must begin with a better set of math standards -- one that is mathematically correct and coherent. Overall, CCSSM meet these criteria in surpassing fashion.

} The CCSSM will undoubtedly be more challenging to all students because, for perhaps the first time, students will be asked to master both procedural and conceptual knowledge and learn each topic in a logical progression.

http://www.huffingtonpost.com/hunghsi-wu/math education_b_1901299.html?view=print&comm_ref=false

To Accelerate, or NotProf. Hung Hsi Wu, UC Berkeley Mathematics

} Skimming over existing materials in order to rush ahead to more advanced topics will no longer be considered good practice.

} Mathematics is by nature hierarchical. Learning it properly requires thorough grounding at each step, and skimming over any topics will only weaken one's ability to tackle more complex material down the road. The weakness usually shows up in students' scientific work in college. This is one reason why many of my colleagues bemoan the practice of acceleration in schools.

http://www.huffingtonpost.com/hunghsi-wu/math education_b_1901299.html?view=print&comm_ref=false

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