Editor’s Note: Another installation in a series of posts evaluating the question: Has Indiana departed from the Common Core State Standards and its attempt to nationalize education in America?
Having been the first state to leave the Common Core, the final draft of Indiana’s new K-12 content standards has been published and it will be brought to the State Board of Education on April 28 – ten days from now – for the final vote. Some reviews of this draft have been already published (e.g., here, here and here) but they focused mostly on the English and Language Arts (ELA) piece. I will focus on its mathematics and I will start with some general observations.
The drafting was done under a serious time pressure. There were only 12 weeks allocated for the standards-writing process that typically takes many months or even years. The writing panels should be commended for significant improvement of its early drafts, yet – as we shall see shortly — the final result is far from satisfying for Indiana, whose prior (pre Common Core) standards were highly praised as the best in the nation.
Previous drafts, particularly in K-8, were almost a carbon copy of the Common Core (CC), with a bunch of Indiana’s own addition on top of them. That created a bloated draft with numerous duplications and excessive number of standards. The final draft has most – but not all – of those duplications removed, and it seems to have a reasonable number of standards, between 21 and 36 per grade. Yet much of the draft is still made of Common Core standards even if their original language has been edited slightly to give the impression of change, and the excessive number of standards was hidden by sticking disparate standards together rather than by removing secondary content from the February bloated draft.
In terms of its language clarity, the final draft is somewhat improved yet still sloppy. It contains standards that are written in an obscure and difficult to comprehend language; standards that are so imprecise that they can be interpreted to mean anything; and standards that contain plain mathematical errors.
When it comes to rigor of expectations, the draft accelerated a couple of early arithmetic skills, yet when it comes to elementary school capstone standards such as fluency with multiplication and division of integers the draft ends up aligned with Common Core’s mediocre expectations and below Indiana’s 2006 standards. Similarly, Algebra 2 content present in Indiana’s 2006 standards has been now excised from Algebra 2 and moved to advanced courses like trigonometry or pre-calculus.
Finally, when it comes to pedagogy embedded in the standards, the situation has not improved much from the early drafts. The preamble to the final draft declares that (original emphasis):
While the standards demonstrate what Hoosier students should know and be able to do in order to be prepared for college and careers, the standards are not instructional practices. The educators and subject matter experts that worked on the standards have taken care to ensure that the standards are free from embedded pedagogy and instructional practices. The standards do not define how teachers should teach.
Unfortunately, this statement is simply untrue. The mathematics standards are infused with pedagogy and dictate – sometimes in painful detail, sometimes with experimental pedagogy – how to teach specific content, rather than leave the pedagogy to schools and classroom teachers.
Specific examples of each of the above mentioned problems are discussed below.
Infusion with Common Core Pedagogy
CC unreasonably and without any justification introduced counting to 100 in kindergarten, and 120 in the first grade. Prior to CC, almost universally, kindergarten counting expectation was 20, and counting to 100 was expected in the first grade. The Indiana draft adopts this baseless – and somewhat meaningless — CC characteristic, presumably as a mark of fealty.
Here is the CC standard 1.OA.6 that I previously used as an example of CC pedagogy:
1.OA.6: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
And here is its final Indiana draft equivalent:
1.CA.1: Demonstrate fluency with addition facts and the corresponding subtraction facts within 20. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). Understand the role of 0 in addition and subtraction.
Let us look carefully at what has happened here. First, Indiana slightly changed the language of the first sentence expecting fluency to 20 whereas CC expected it only to 10 (and less than fluency to 20). Nice but quite meaningless, as in the second grade both CC and the final draft already expect identical ability to fluently add and subtract within 100. The second part is identical, where CC pedagogy is carried intact into the final draft. Finally, the last sentence about the role of zero is carried over from the 2006 Indiana standards, nicely illustrating how the final draft achieved its seemingly-reasonable number of standards – the draft simply lumps distinct standards together to make them into one.
Here is the grade six ratio standard infused with pedagogy and copying Common Core prescriptiveness:
CC 6.RP.3: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
Draft 6.NS.10: Use reasoning involving rates and ratios to model real-world and other mathematical problems (e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations).
Fluency with arithmetic of integers and fractions is frequently – and correctly – identified as the focus of elementary grades. Like CC, Indiana’s final draft expects fluency with addition and subtraction of integers by grade four, multiplication by grade five, and division by grade six. In contrast, Singapore is done with addition and subtraction by grade three and with multiplication and division by grade four. The 2006 Indiana standards expected both multiplication and division of integers to be completed by grade five, a year ahead of CC. Here is a small example of how the draft’s rigor and pedagogy compare internationally (grade three):
Draft 3.C.4: Interpret whole-number quotients of whole numbers (e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each).
Singapore, gr. 3: multiplication and division of numbers up to 3 digits by 1 digit (with remainder)
When it comes to fractions, the final draft defers fluency with the four arithmetic operations on fractions to grade six like Common Core. In contrast, the 2006 Indiana standards (and Singapore) expect it a year earlier, in grade five.
Studying circles is postponed to grade seven from grade six in 2006. Constructions with straight-edge and compass are postponed to high school from grade eight in 2006. Introducing and using the Pythagorean theorem is postponed to grade eight from grade seven in 2006.
Algebra 2 in the final draft excludes conic sections and doesn’t expect students to work with logarithms beyond just their definition – content that was present in the 2006 Indiana standards, and this content has been pushed in this draft to trigonometry and pre-calculus. Inductive proofs were completely eliminated from this draft.
Sloppy and/or Erroneous Language
There are too numerous cases where the draft’s language is sloppy and unclear to list them all here. A small selection is offered. · In numerous places the draft replaced CC language of arithmetic fluency “using the standard algorithm” with “using a standard algorithmic approach.” This seemingly innocuous alteration reflects a radical change undermining the expectation of students ever learning efficient and universally practiced standard arithmetic procedures for computation. While there is effectively a single “standard algorithm” for multiplication, “a standard algorithmic approach” includes, for example, also the inefficient diagonal lattice multiplication. Similarly, while we all know the standard vertical algorithm for addition, “a standard algorithmic approach” for addition may also include the abacus or systematic counting on one’s fingers and toes.
· K.CA.3: Use objects, drawings, etc., to decompose numbers less than or equal to 10 into pairs in more than one way, and record each decomposition with a drawing or an equation (e.g., 5 = 2 + 3 and 5 = 4 + 1). [In Kindergarten, students should see equations and be encouraged to trace them, however, writing equations is not required.]
How can one ”record each decomposition with … an equation” if “writing equations is not required”?
· 7.C.8: Solve real-world problems with rational numbers by using one or two operations.
Unclear if the intent is solving an expression with one or two arithmetic operations, or solving one- or two-step problems.
· 8.AF.8: Understand that solutions to a system of two linear equations correspond to points of intersection of their graphs because points of intersection satisfy both equations simultaneously.
A system of two linear equations typically has a single solution (two lines do not intersect at multiple points), while the standards implies multiple points of intersection.
Unfortunately, the selection above is just a sample of problems with the final draft.
Until recently, Indiana standards were among the best in the nation. They were concise, clear, demanding, and mostly bereft of pedagogy. When Common Core was adopted, Indiana traded its excellent standards with mediocre, pedagogy-infused standards, written in Washington, DC. It is commendable that Indiana is the first state to realize the mediocrity of Common Core and abandon it, yet the proposed draft is far below the excellent quality Indiana has been used to.
There is an easy and obvious solution to this dilemma. Indiana should temporarily re-adopt its excellent 2006 standards. It already has a test aligned to those standards, and the only issue is to have those 2006 standards declared “college-ready” to preserve Indiana’s NCLB flexibility waiver. This should be an easy task given that the 2006 standards are of higher quality and higher rigor than the current draft, which has already been “blessed” as college-ready by Indiana’s universities and colleges.
Once the 2006 standards are re-adopted, Indiana should embark on a deliberate and unhurried process of improving its already-excellent standards.