I am a member of a team of math educators in Vermont that is working with the Vermont Agency of Education to help spread the word around the state about math education initiatives. Vermont is a small, rural state and teachers are often isolated. The Agency of Education employs 2 (actually 1.5) very dedicated educators to oversee mathematics for the whole state. Tracy Watterson oversees K – 5 and Lara White oversees 6 – 12, but for only half of her time. Her other time is spent in another .5 Agency of Ed position. This year, the 6 – 12 team is hosting bi-monthly meetings in 7 regions around the state to bring information directly to the teachers rather than the teachers having to come to a central location. We’re calling these gatherings Math Morsels … we bring food morsels as an enticement! We meet from 4 to 6 PM, so teachers do not have to take time off during the school day. We are in the midst of Round 2. The focus of this round has been to familiarize teachers with the Smarter Balanced (SBAC) assessment that will be administered for the first time this spring to Vermont students.

The SBAC is new to everyone, so this first year will most likely be full of much consternation and headaches for educators and students. As in any new thing, my advice to teachers is to go in as informed and prepared as you and your students can be, and accept the inevitable fact that issues will arise that you didn’t even consider. Don’t beat up on yourself or your students. Learn from the experience and make changes to improve the process over time. Also, There will be complaints about the sorry state of education and how awful the Common Core is and there will be an appeal to go back to the basics, whatever they are/were.

The “basics” have changed. The predominant metric that we have traditionally used to assess students is the ability to get a correct answer quickly. We have many tools that can give us an “answer”. The tools that our students (and we) lack are the habits of mind that are needed to persevere and solve non-routine problems. These habits are encapsulated in the 8 Practice Standards:

# 1 Make sense of problems and persevere in solving them.

# 2 Reason abstractly and quantitatively.

# 3 Construct viable arguments and critique the reasoning of others.

# 4 Model with mathematics.

# 5 Use appropriate tools strategically.

# 6 Attend to precision.

# 7 Look for and make use of structure.

# 8 Look for and express regularity in repeated reasoning.

The most important change that we need to make as educators in order to prepare our students for success on the SBAC is not the content that we teach. The most important instructional change we can make is to teach our students how to persevere in solving a problem.

As we looked through the test resources, it became clear that there are some problems that are asked in very different ways than we and the students are used to. In order to correctly answer the questions, students need to be creative problem solvers. Here’s an example of a 6th grade task. Students are shown a square and told that the area of the square is 324 square units. The question asks for the length of a side.

This is a pretty straightforward question. The concept being assessed is that in a square the area is found by multiplying a side length by itself. However, while a calculator is provided for many problems, there was not one available for this problem. (Each problem is worked on a computer with one problem per page.) A teacher that I was sitting next to and I were surprised that there was no calculator, since we normally don’t expect students to have the square root of 324 memorized. In fact, I myself didn’t know it off the top of my head. However, I had some strategies that I have developed over the years that could lead me to the answer pretty quickly.

This type of question has embedded within it several of the Common Core Math Practice Standards:

# 1 Make sense of problems and persevere in solving them.

# 2 Reason abstractly and quantitatively.

# 7 Look for and make use of structure.

Students could of course guess and check 1 x 1, 2 x 2, until they found it, This would be an example of # 1, making sense (knowing that a number times itself needs to have a product of 324). It would also be an ad nauseum example of perseverance.

However, if the student added the abstract and quantitative reasoning expected by # 2, they might start with larger easy numbers, such as 10 x 10 and 20 x 20 to get a rough idea of the magnitude of the number they were looking for. Once they looked at 100 and 400, they knew that they number they were looking for was closer to 400 and could guess and check 19 x 19, 18 x 18, and voila, they are there.

If they did the above, but instead of finding 19 x 19, and 18 x 18, they looked at the structure of 19 and realized that the ones place of 19 x 19 would be 1 (since 9 x 9 = 81), they wouldn’t need to do that multiplication. They could move to 18 x 18, see that the ones place would be 4 (since 8 x 8 = 64), then test out 18 x 18, which would prove that a side length of 18 is the correct answer.

Another approach that was mentioned at the meeting that I had not thought of was to factor 324 and group the factors so that there were two sets of the same factors multiplied together. 324 = 2 x 162, which is 2 x 2 x 81, which is 2 x 2 x 9 x 9, s0 324 could be expressed as (2 x 9) x (2 x 9), which is 18 x 18. This is another example of using Practice Standards # 1, 2 and 7.

The problem was not an extremely difficult problem. It just had a number that a student might not have memorized as a perfect square. A student who had been given plenty of opportunities to make sense of problems and persevere, who were comfortable with the structure of numbers and operations would have the tools to work through this problem fairly quickly. A student who expected a quick answer or a quick tool to figure it out would most likely skip it and move on.

The Practice Standards provide students with habits of mind that will serve them well as they work through non-routine problems. Let’s face it. All problems that our students will face in life are non-routine. Let’s prepare our students for a future in which they will persevere and expect that any problem worth solving may take some thought and can be solved using multiple strategies. Doing this will mean changing the way we instruct students to approach problems. It’s not enough to have the 8 Practice Standards posted on the wall. We must refer to them often, use them in our own practice, and expect students to do the same.

My questions are:

Would a problem like the one shown be graded by computer?

Are the students expected to explain their answer?

If so, does the explanation carry any marks?

How would the different explanations be assessed?

What about the kid that guessed right first time?

I don’t want answers ! ! !