Blame the teachers. That’s the advice from PWCS for anyone with concerns about the Math program.

I think PWCS is wrong. I don’t think our teachers are at fault for the problems that continue in our elementary level math program. I don’t think our elementary level teachers have the autonomy the school division says they do because they have to teach to the test. Not the SOLs, the CFAs.

**Teaching to the CFA
**

Staff asserted that retired PWCS teachers developed the end-of-unit assessments, called Common Formative Assessments or CFAs, that PWC students are given throughout the year. These assessments include multiple choice and free response or open ended questions and are aligned with the level and type of questioning found in the new VA SOLs. The assessments were developed to test knowledge and skills attained at the end of each unit and to provide data to teachers so that they can tailor instruction to address any deficiencies in their students’ knowledge base. *{note: Free response or open ended means if they’re asked to solve 6 x 8 , rather than being given four options to choose from, students must actually type in the correct answer. Sometimes free response answers are written explanations of operations or drawings.}*

I don’t have any major concerns with the CFAs. While I think PWCS may have taken the whole data driven instruction thing too far and am concerned that the CFAs focus too heavily on preparedness for the SOL exams as opposed to comprehension and fluency, I do agree that there are some benefits to periodically testing students to ensure that they’re learning what they need to learn.

My concern is with how the CFAs have been scheduled and how that governs the sequence of instruction in our classrooms.

Under questioning, staff disclosed that the CFAs were based on the sequencing of lessons and units from Math Investigations. Staff asserted that this wasn’t an issue because PWCS teaches the curriculum and not a textbook.

Here’s why I believe it’s an issue.

One of the major new topics in 4th grade is multi-digit multiplication. Like so many things in math, multi-digit multiplication is something that relies on previously learned skills. You start by learning multiplication facts (5 x 7), then learn how to multiply larger numbers by a 1 digit number (543 x 7), then then learn how the multiply larger numbers by 10’s (543 x 20), and then pull it all together and learn how to multiply two multi-digit numbers (543 x 21). It’s a sequential process where each step is dependent on mastering the previous step. If you stumble on one step, you’re going to struggle to reach the next step.

Math Investigations uses an instructional approach called spiraling, which means you start one topic, get part way through it, leave it to do something different, and then come back and continue with the original topic some time later. In Math Investigations, multi-digit multiplication is taught in two units that are separated by a 3 week unit on data analysis. Math Investigations students do the first unit on multiplication and division which gets them to the step where they’re multiplying larger numbers by a 1-digit number, then then do a 3 week unit on organizing and interpreting data in bar charts and line graphs, and then the come back to the second unit on multiplication and learn multi-digit multiplication.

When the CFAs were written, they were designed to follow that sequence of instruction, so the first CFA would be completed after the first unit on multiplication and division and would cover that content, the second CFA would be completed after the unit on data analysis and would that content, and the third CFA would be completed after the second unit on multiplication would cover that content. This allowed teachers to follow the Investigations materials in order starting at the beginning and moving on to the end.

Even though the CFAs followed the sequence of lessons in the order suggested by Math Investigations, there were no concerns about that undermining our curriculum as our curriculum was the VA SOLs and the CFAs were designed by our teachers to test the knowledge and skills they believed students needed pass the SOL exams and move ahead.

Math Connects, our new elementary math textbook, doesn’t follow the same sequence of lessons that Investigations followed. Connects students learn multi-digit multiplication in two chapters (units) that follow one another in sequential order. Chapter 4 covers multiplying larger numbers by a 1-digit number and 10’s and Chapter 5 covers multi-digit multiplication. Organizing and interpreting data in bar charts and line graphs isn’t covered until Chapter 7. Chapter 6 is long division.

The schedule for this year is the same schedule followed when Math Investigations was the primary instructional resource. The schedule calls for 4th grade students to do a unit on multiplying larger numbers by a 1-digit number and 10’s followed by the CFA for that content, then do a unit on organizing and interpreting data in bar charts and line graphs followed by the CFA for that unit, and then do the final unit on multi-digit multiplication followed by the CFA for that content. That means our 4th grade students do Chapter 4, then parts of Chapter 7, and then Chapter 5.

This may not seem that bad, except the lessons in Chapter 7 are based on students having completed Chapters 4, 5, and 6. Many of the problems, particularly the homework, require double digit multiplication and long division to solve, but our students haven’t developed those skills yet because they haven’t done those lessons.

PWCS staff claimed in the presentation that this hopping all over the place was necessary to meet the PWC Curriculum, but, as I noted in Part (2) of the Blame the Teachers series, the PWC Mathematics Curriculum is the VA SOLs and there is nothing in the VA SOLs that requires lessons to be taught in the order PWCS requires.

This same hopping all over the place sequencing of lessons is followed in 3rd and 5th grades. To make things even more complicated, the lessons within Chapters aren’t completed in sequential order and are frequently replaced with materials from Investigations and other texts. Only 47% of the lessons on multi-digit multiplication come from Connects. The rest are from Investigations and other sources (I’ll expand on that in more detail in Part (4) of Blame the Teachers). And yes, Connects did and does meet VA DOE SOL standards for multi-digit multiplication.

Per PWCS staff, teachers have the autonomy and authority to pace lessons however they want to meet their students needs. What wasn’t mentioned was that teachers MUST give the CFAs as close as possible to the date specified in the pacing guides. Our teachers have to teach material to their students based in what will be on the CFA, which means they have to sequence lessons as the pacing guides direct. So you get a three week visit to data analysis land when you’re in the middle of building to multi-digit multiplication.

This is how PWCS intentionally designed things.

Just to clarify, because I know some folks will have their panties in a wad over this, I’m not saying that we mindlessly teach the pages in the textbook like robots. PWCS could still use the teacher developed CFAs with the same questions and same emphasis on whatever it is our teachers have decided to emphasize, but paced to align with the sequence of instruction Connects follows. PWCS would have the same curriculum, same content emphasis, and same “supplemental” materials, but would actually be teaching the lessons in the order they were meant to be taught. Because our teachers actually be using the text in the order lessons were meant to be taught, our students would know what page they were doing in the book, parents would be able to follow along and actually support their children, and students wouldn’t be faced with homework or classroom exercises that were based on skills they hadn’t attained.

To demonstrate how little would have to change, here’s how PWCS could have scheduled lessons and CFAs, had they been so motivated.

**How PWCS Could Have Done It**

For 4th grade, the schedule could have been as follows, with Chapter 3 moved to just before the SOLs:

- Chapter 1- Place Value, Addition, Subtraction
- CFA on Place Value, Addition, Subtraction

- Chapter 2 – Multiplication and Division Operations and Fact Review
- Chapter 4 – Multiply by 1-digit number
- CFA on Multiplication and Division 1

- Chapter 5 – Multiply Multi-Digit Numbers
- Chapter 6 – Division
- CFA on Multiplication and Division 2

- Chapter 7 – Organize and Display Data & Probability
- CFA on Data Analysis and portion of CFA on Modeling Fractions and Decimals and Probability that relates to probability

- Chapter 8 – Decimals and Fractions
- Portion of CFA on Modeling Fractions and Decimals and Probability that relates to modeling fractions and decimals

- Chapter 9 – Add and Subtract Fractions and Decimals
- CFA on Fraction and Decimal computation

- Chapter 10 – Identify and Describe Geometric Shapes
- Chapter 11 – Spatial Reasoning (parallel, line segments, points on a grid, transformations)
- CFA on Geometry

- Chapter 12 – Measure Length
- Chapter 13 – Measure Capacity, Weight, and Mass
- CFA on Measurement

- Chapter 3 – Algebraic Properties
- CFA on Patterns, Functions, and Algebra

Under this scenario only one CFA would have to change, and even then the change wouldn’t be content based but logistical as you’d be removing the probability questions from the CFA on Modeling Fractions and Decimals and Probability and adding them to the CFA on Data Analysis.

The test questions and content emphasis wouldn’t be altered at all, nor would the curriculum change. We’d just be aligning the CFAs with the sequence of instruction followed by the textbook our school board approved. You may think that’s a minor point, but it’s not.

When the school board adopted Math Investigations we Mad Math folks were told repeatedly that it was just a textbook. We Mad Math folks argued that it was more, and we were told we were off our rockers. In the last few years officials with PWCS have openly admitted that the school board adopted an inquiry-based, spiraling approach to instruction when it adopted Investigations. Investigations wasn’t just a textbook, it was much more, just like we’d said years earlier. Adopting “more” may not have been overt and the school board members may not have realized that that’s what they did when the adopted Investigations, but that was the message the Math Department and school division took away from it and is how they’ve operated since then.

That changed when the school board adopted Math Connects for K – 5 instruction. Connects is a mix of inquiry-based and traditional methods that doesn’t spiral instruction.

To understand why that’s such an important point, check out part (4) of our Blame the Teacher series.

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