A Walk in the PARCC – Part 1: Math

Guest Post by Stig Leschly, CEO of Match Education

Massachusetts is moving to the new national standards (Common Core) and related tests (PARCC). At least so it seems.  The politics of it all are complex and hard to predict.  They’ll play out over the next few years.

This post is not about the politics, though.  It’s about the substance of it all—about how the new standards and tests look, here on the ground, to our students and teachers.

This post is the first of two.  It addresses the move from MCAS to PARCC in math.

A second post, later this week, will cover the move from MCAS to PARCC in ELA.

Here at Match, we like PARCC-Math over its predecessor, the MCAS-Math.  Here are some of the reasons why, each of which I’ll cover in detail in this post:

  1. PARCC-Math mitigates the role of luck.
  2. PARCC-Math, more than MCAS-Math, requires students to understand math conceptually.
  3. PARCC-Math, more than MCAS-Math, requires students to provide detailed explanations for their solutions.
  4. PARCC-Math open response questions, more than their MCAS analogs, demand that students identify relevant information and solve complex, multi-step problems.

1.  PARCC-Math mitigates the role of luck

Guessing is invariably part of test taking, but less so with PARCC-Math than with MCAS-Math.

A typical MCAS-Math multiple-choice question asks a student to select one of four correct answers.  For example, consider this order-of-operations question from the 5th grade MCAS-Math:

A student with no knowledge of order of operations has a 25% chance of guessing the correct answer.

PARCC-Math greatly reduces a student’s odds of lucking into the correct answer.  PARCC-Math asks fewer straight multiple-choice questions and, instead, favors open-response questions.  Where PARCC-Math seeks a simple numerical answer, it often asks students to fill out an answer grid, rather than offering a series of possible answers.

Here is an example of an order-of-operations question from 5th grade PARCC-Math, one that involves a “grid” answer key:

Even when PARCC does ask standard multiple-choice questions, it often asks the student to select “each” correct answer, without specifying how many correct answers are present in the answer line up.

Here is an example of a 7th grade PARCC-Math multiple-choice question with multiple correct answers:

The question above has two correct answers (A and E).  The odds of randomly guessing A and E are miniscule (about 3%).  Students cannot easily luck into showing mastery on PARCC-Math.

2.  PARCC-Math, more that MCAS-Math, requires students to understand math conceptually

Consider the following 7th grade MCAS-Math question on volume:

To answer this question, a student needs simply to apply the formula for volume (L x W x H) but does not need to understand volume conceptually (i.e. be able to spot a volume question that is not overtly described as such).

Moreover, in the question above, the values for the formula are obvious.  A student does not need to be selective in identifying relevant data to input into the volume formula.

For contrast, consider this 7th grade PARCC-Math question on volume:

In this PARCC-Math volume question, a student still needs to know and be able to apply the basic formula for volume of a rectangular prism.

But notice the additional challenges involved in the PARCC-Math version.

First, students have to arrive independently at the values for L, W, and H in the volume formula.   They are not given or obvious.  That challenge alone will stump students who have only a superficial understanding of volume.

Second, students have to know the meaning of a “right rectangular prism” in order to downsize the block correctly by 20 units, as directed in the third bullet of the question.

And third, students have to show their work and logic as they progress through the problem.

In short, students truly have to understand volume as a phenomenon in order to pass this PARCC-Math question.  Mastering the formula for volume alone is not enough.

3.  PARCC-Math, more than MCAS-Math, requires students to provide detailed explanations for their solutions

Historically, MCAS-Math rarely asked students to explain their choice of math algorithm.  For example, consider this 5th grade MCAS-Math problem on fractions:

By comparison, consider this 5th grade PARCC-Math question, also on the addition and subtraction of fractions:

To get full credit on this PARCC-Math question, a student obviously needs to be able to subtract 1 3/4 from 3 2/4.

But she also needs to understand this procedure as the one that is called for in the word problem and – for full credit – to go further and explain how the protagonist mishandled the problem.

The question, far more than a conventional MCAS-Math question, demands fluency with fractions.

4.  PARCC-Math open response questions, more than their MCAS analogs, demand that students sort for relevant information and solve complex, multi-step problems

The most challenging MCAS-Math questions are “open response” questions.  These questions typically ask a string of 2 or 3 questions, sequenced in a way that guides a student through a problem.

Here is an example of an MCAS-Math open response question in 5th grade.  It deals mainly with multiplication and division:

This question is, in our opinion, a solid test of a student’s mastery of multiplication and division.  But, it could be a lot more demanding.

In particular, notice how a question is inserted immediately following the information relevant to that question.  In this way, the question guides students.

By contrast, PARCC-Math open-response questions tend to present a full and often lengthy word problem and then, at the end, ask a series of questions that require students to sort for and manipulate relevant information.

Here is a 5th grade PARCC-Math open-response question that also tests multiplication and division.  It is much harder, as you will see:

On this PARCC-Math question students have to take seven or eight steps to reach a solution, whereas the MCAS-Math analog above required only three or four.   And students have to search intensively for information relevant to each step of the solution.  The question truly tests students’ ability to parse information in context, to think conceptually, and to discern an efficient path to a solution.

This PARCC-Math question also involves numbers that are plainly more difficult to multiply and divide than the numbers on the MCAS-Math questions.   The last step on the PARCC-Math question involves dividing 1,491 by 24, finding the answer as 62 with remainder 3, and concluding as a result that at least 63 cases of water are needed.  By comparison, the most complex operation on the MCAS-Math question is to divide 180 by 18.


In all, we like PARCC-Math here at Match.

The new tests and the Common Core standards offer a more rigorous approach to math skills and knowledge than their predecessors, in our view.  They prepare our students more clearly and from an earlier age for the challenges of advanced math (algebra, geometry, trigonometry, and calculus) that they will face at our high school and in college.

PARCC-Math is a challenge worthy of our students and of our teachers.

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The following people gave input in writing this post: Ryan Holmes, Jamie Morrison, Ray Schleck, Meredith Segal, Jennifer Spencer, Kyla Spindler, Emily Stainer

3 thoughts on “A Walk in the PARCC – Part 1: Math”

  1. I for the most part agree with your assessment of the PARCC math tests. I think it’s important to note that these examples come from the paper-based PARCC, which many districts used this year as they transition to the Common Core, while it seems likely that all, or almost all, districts will take a computer-based test next year. This doesn’t change the substance of your comments, but it does mitigate the role of luck even more — some questions involve typing in numbers, or dragging a line and selecting where to shade to show an inequality, or filling in the blanks. There are a number of places where the paper-based PARCC was revealed to be confusing and unnecessarily complicated compared to the computer test. For instance, one question on the Algebra I test asks students to find a rate. On the computer-based test, students select the number from a drop-down menu, and then select the units from a second drop-down menu — a great way of assessing the relevant skill of what units to apply to a rate without letting that get in the way of or cloud the computation. However, on the paper-based PARCC (practice test) this was a multiple choice, select-all-that-apply question. Answers A-D were numbers, and answers E-H were units. It was a confusing way to word the problem, and I think is a particular disadvantage for students with disabilities that clouds the standard being assessed. There are a number of similar interactive tools that have been developed for the computer-based test that I think are excellent ways to focus students’ attention on the concepts at hand, and less on the mechanics of the question.

    Your conclusion is interesting —

    "The new tests and the Common Core standards offer a more rigorous approach to math skills and knowledge than their predecessors, in our view. They prepare our students more clearly and from an earlier age for the challenges of advanced math (algebra, geometry, trigonometry, and calculus) that they will face at our high school and in college."

    Is your argument that the purpose of math is to learn more math? I would question that. In my view, the purpose of learning math is to practice looking for patterns and generalizing, reasoning inductively and deductively, representing solutions clearly, justifying answers, and validating conclusions. These are the skills that I want my students to be assessed on — to see whether they have a deep base of knowledge, and can apply that knowledge in a variety of ways authentic to the practice of mathematics. That is a test worthy of my students, but taking a test to make sure my students are ready for the next test sounds like circular reasoning to me.

    One thing that worries me about the PARCC tests is the continued emphasis on rigorous procedures that don’t support higher math. I am not one to argue that since our students can "look anything up on Google", basic facts are irrelevant. Fluency — built from conceptual understanding — is the basis of deep knowledge that can be applied in a variety of ways. But I don’t think we should highlight asking students to 1,491 by 24 as a success. I believe that students should be able to do it — but 180 divided by 18 is also difficult for most fifth graders, and placing the emphasis on 180 divided by 18 provides an incentive to teach place value and varied strategies for multiplication, rather than increasing practice with calculation that students will not be asked to do by hand again, and that does little to reinforce meaningful higher math. These are rigorous, multi-part questions that require students to hold a great deal of information in working memory. Adding to that load a procedural question seemingly designed to tax working memory as much as possible washes out the chance of assessing both skills in a meaningful way. As a teacher, I want to know if my students can do a difficult procedural question — but having given my students plenty of problem-solving tasks, I have learned that one deep pit of difficult procedural work can distract from reasoning to the point of making the task worthless. These are separate parts of math, and should be assessed separately.

    You also didn’t mention the Standards for Mathematical Practice — the motivation for much of this change. The Common Core standards aren’t as different from our old standards as some might think — and they certainly don’t say "ask questions on the same topics as before, but make them harder". Instead, they choose to focus our attention on the eight Standards for Mathematical Practice, where many believe the true emphasis of Common Core math should be. PARCC has chosen to focus on assessing three of these — construct viable arguments and critique the reasoning of others, model with mathematics, and attend to precision. Several of the questions you chose are meant to assess whether students can model with mathematics, and this is likely the toughest of these practices. This is where I think the PARCC test falls the most short. If students are to truly model with mathematics, they must be asked to do things that go beyond the scope of a standardized test. They must be able to decide what information is necessary to solve a problem — that doesn’t mean throw a ton of information into a question so it’s confusing for adults, it means we should ask students to imagine a stack of every penny ever created — and ask them how high that stack would be. What would you need to know to answer that question? How would you measure it? What math would you use? Students must be able to compute, and present their result — but they most also acknowledge the error in doing math in the world, and consider how to account for it, or when to go back and calculate again when their answer is inconsistent with the goal of the problem.

    These skills are impossible to assess on a traditional standardized test, although PARCC is trying, and I think is getting closer than tests have before. But I worry that, as we move deeper into the world of high-stakes testing, with more days of tests and more strings attached for students and teachers, we remove the incentive to meet the true goals of Common Core mathematics. Comparing a new test to an old test does not mention where mathematics teaching could go — and many others right now are working on exactly that. The test is better, but the test cannot be the be-all-end-all of math teaching, and that is what I am worried about as test season wraps up this year.

  2. I love Dylan’s comment.

    I am curious how your students do on the multi-select questions in practice, and how you are helping them to tackle these problems in an efficient and effective way.

    I am finding that they are massively challenging to my students, across all subject areas. I guess I understand the need to reduce kids guessing their way to a proficient score, but I am opposed to the right/wrong nature of these questions. It seems to me that there should be some nuance in this type of question. For example, let’s say a question asks for a student to find all the representations of proportional relationships in the answer choices. Let’s say there are 6 choices and 3 are right. Isn’t just as arbitrary to say that the student gets no credit if they find 2 out of 3, than it is to say they get credit if they randomly guess? It seems to me that multi-select questions have the potential to be fantastic diagnostic tools to identify which aspects of an idea students have mastered and which they haven’t, but it can be so frustrating for teachers and students to get those questions wrong at such a high rate. I find this true for even many of my strongest students – and this is true across our network. On every test on which I put a multi-select question – science, math, you name it, these are ALWAYS the hardest questions regardless of content. I wonder, then, if my kids just need more practice, or there is something flawed about this style of question. PARCC is the first standardized assessment I’ve ever seen that uses this question strategy, and it is surprising to me how often this is their "go-to" MC choice.

  3. How do you find students with weak reading and comprehension skills cope with this sort of approach, especially for something like the last example?

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