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The California Mathematics Standards. The California Standards come in two flavors. General Standards Green dot standards. In California the green dot standards are the focus of most instruction since these topics comprise 85% of the state exams in grades 2 – 7.

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The california standards come in two flavors
The California Standards come in two flavors

  • General Standards

  • Green dot standards



California intervention
California Intervention instruction since these topics comprise 85% of the state exams in grades 2 – 7.

  • The new California Framework has the same standards as the 1998 Framework.

  • But it also has a new appendix on intervention programs.


Algebra readiness
Algebra Readiness instruction since these topics comprise 85% of the state exams in grades 2 – 7.

  • For at risk students there is also a discussion of the requirements for algebra preparedness:


The original material
The original material instruction since these topics comprise 85% of the state exams in grades 2 – 7.

  • Unfortunately, politics muddied the water, and the exposition is not quite what H.-H. Wu and I provided.


The nctm focus topics

The NCTM Focus Topics instruction since these topics comprise 85% of the state exams in grades 2 – 7.


The focus topics
The Focus Topics instruction since these topics comprise 85% of the state exams in grades 2 – 7.


The focus topics1
The Focus Topics instruction since these topics comprise 85% of the state exams in grades 2 – 7.


The role of the focus topics
The Role of the Focus Topics instruction since these topics comprise 85% of the state exams in grades 2 – 7.

  • NCTM regards the Focus Topics as a description of how the key elements in an effective curriculum in mathematics should look.

  • But comparing the Focus Topics with most state standards reveals a very poor fit.

  • An exception is California.

    • The Green Dot standards closely align with the Focus Topics. And the emphasis is also about right

    • NCTM recommends 60 – 80% of instruction time be devoted to the focus topics, while 85% of the California assessments in mathematics are on these standards.


Focus topics numbers
Focus Topics – Numbers instruction since these topics comprise 85% of the state exams in grades 2 – 7.


Focus topics fractions
Focus Topics Fractions instruction since these topics comprise 85% of the state exams in grades 2 – 7.


Focus topics fractions ratios
Focus Topics Fractions, Ratios instruction since these topics comprise 85% of the state exams in grades 2 – 7.


Focus topics algebra data
Focus Topics, Algebra, Data instruction since these topics comprise 85% of the state exams in grades 2 – 7.


  • There are more topics among the green dot standards instruction since these topics comprise 85% of the state exams in grades 2 – 7.

  • There are minor differences in the grade levels for different topics between the two documents

  • But there are no substantive differences in the sequencing of the common topics



A consensus is emerging

A Consensus is Emerging subjects that should be given the vast majority of instruction time in K – 8.

In fact, standards designed in this way cut the need for intervention and support maximal achievement for all students


The programs in the top achieving countries follow these principles

The Programs in the top achieving countries follow these principles.

And their data comprises literally billions of people


We now look at some mathematical issues that should be part of the considerations in developing standards.


Many, if not most, standards of the considerations in developing standards.need illustrative problems and these problems need to be both mathematically accurate and carefully selected


First Page of the considerations in developing standards.

Kindergarten

Standards:

California


The same holds for test questions of the considerations in developing standards.

and problems in student texts


Problem from Current U.S. Text of the considerations in developing standards.

Aconcagua is 6962 meters high. Mount McKinley is 6194 meters high. Which will take longer to climb?

This was the only problem a very bright third grader “missed” in the best elementary school in his district.


Naep the nation s report card
NAEP – The Nation’s Report Card of the considerations in developing standards.

  • Last year, The Brookings Institute asked me to review the algebra questions on the NAEP

  • Of the 41 eighth grade NAEP algebra problems provided, 8 were incorrect and one was meaningless

  • Moreover about 10 of the correct problems were just questions about vocabulary, not mathematics



From the naep
From the NAEP four others were essentially vocabulary, only one could be judged mildly challenging at fourth grade level


Now let s look at examples from high achieving programs

Now Let’s Look at examples from High Achieving Programs four others were essentially vocabulary, only one could be judged mildly challenging at fourth grade level


First Grade four others were essentially vocabulary, only one could be judged mildly challenging at fourth grade level

Russia


Russia: Grade 2 four others were essentially vocabulary, only one could be judged mildly challenging at fourth grade level

Distributive

Law


A third grade lesson from singapore
A Third Grade Lesson from Singapore four others were essentially vocabulary, only one could be judged mildly challenging at fourth grade level


A Problem From Russia: Grade 4 four others were essentially vocabulary, only one could be judged mildly challenging at fourth grade level

A lone goose was flying in the opposite direction from a gaggle of geese.

He cried: “Hello, 100 geese!”

The leader of the flock answered: “We aren’t 100! If you take twice our

number and add half our number, and half of half of our number, and

finally add you, the result is 100, but … well, you figure it out.”

How many geese in the gaggle?

99/(11/4) = 36


At the end of the day this level of instruction is what i think all of us want for our kids

At the end of the day this level of instruction is what I think all of us want for our kids

And top flight, carefully designed standards are the basic step in achieving this outcome


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