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. They were selected by mathematicians, teachers and math educators as the key topics students needed to know

California Intervention The new California Framework has the same standards as the 1998 Framework. But it also has a new appendix on intervention programs.

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

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

The NCTM Focus Topics

The Focus Topics

The Role of the Focus Topics 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 Fractions

Focus Topics Fractions, Ratios

Focus Topics, Algebra, Data

There are more topics among the green dot standards 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

And no disagreement that the focus topics are the core subjects that should be given the vast majority of instruction time in K – 8. They should be learned in depth by students as they provide the foundation for all the math and science that follows

A Consensus is Emerging 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. 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 need illustrative problems and these problems need to be both mathematically accurate and carefully selected

First Page Kindergarten Standards: California

The same holds for test questions and problems in student texts

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. Problem from Current U.S. Text

Problem From WA Practice Exit Exam 6 of first 8 were mathematically incorrect.

NAEP – The Nation’s Report Card 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

Of the 22 grade 4 questions provided, four were incorrect, four others were essentially vocabulary, only one could be judged mildly challenging at fourth grade level

From the NAEP

From NAEP

Now Let’s Look at examples from High Achieving Programs

First Grade Russia

First Grade: Russia: Rest of Lesson

Russia: Grade 2 Distributive Law

Russia: Grade 2 Rest of Lesson

A Third Grade Lesson from Singapore

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? A Problem From Russia: Grade 4 99/(11/4) = 36

Fifth Grade Singapore: Pre Algebra

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