Monthly Archives: January 2014

Asking questions that help students think algebraically

Over the past two weeks, I’ve been working with both students and teachers on algebra tasks. One of the things we have been doing is writing rules for different geometric patterns.

For me, identifying key questions that will guide students towards meeting the learning target for the day is one of the most important things I do when planning a lesson.

I find that it can be easy to do really cool activities that get students engaged and have great potential but if the questions we ask don’t pull out the mathematics, kids get excited about math class but don’t learn much. The questions asked as the students work need to guide them to see the connections and pull the mathematics out of the situation (which may be obvious to us but not to them).

I think sometimes the resistance to inquiry based approaches is because if students aren’t carefully guided to see the mathematical connections, it’s no better (and maybe even worse) than simply telling them a procedure and having them practice a bunch of problems.

When working on my algebra lesson, I went to researcher Mark Driscoll’s book, Fostering Algebraic Thinking: A guide for teachers, Grades 6-10. In it, there are various questions that can be used to foster different types of algebraic reasoning. Below are some of the questions he suggests using when students are engaged in writing rules to represent functions:

Questions for building rules to represent functions

  • How are things changing?
  • Is there information here that lets me predict what’s going to happen?
  • What steps am I doing over and over?
  • How can I describe the steps without using specific inputs?
  • What if I do the same thing with different numbers? What still holds true?  What changes?
  • Can I write down a rule that will do this job once and for all?
  • Why does the rule work the way it does?
  • Does my rule work for all cases?
  • Now that I have an equation, how do the numbers in the equation relate to the problem context?

I used these questions to help assist small groups when they were stuck as well as to guide the whole class discussion. They helped students see the connection between the pattern they were drawing and the algebraic rule that could be written to describe it.

Want to know more?  Check out: Fostering Algebraic Thinking: A guide for teachers, Grades 6-10 by Mark Driscoll

Learning Targets


The first step in thinking about how to provide students with frequent feedback is having students identify their learning targets. Not just the goals for the unit or year, but also the daily or weekly targets that are going to get them there.

As I’ve mentioned before, I’m training for a half-marathon. The event is April 26th. My goal is run all 13.1 miles before they close the course (which I think is after 4 hours) without any permanent damage to myself.

In order to do this, I set weekly goals. For example, this week, my goal is to complete one 5-mile run and two 30-minute runs. For me, that’s much more manageable than thinking about running 13.1 miles.

I track my progress using an app on my phone and I know right after each run if I hit my target. At the end of the week, I know if I’m on track or not and I set my goal for the next week.

How does this relate to math class?

Do your students know what their goal for class was for the day? For the week? For the unit? If they don’t know the goal, they can’t begin to monitor their progress.

Often, we, as teachers, know what the plan is.  We have an objective in our lesson plan or an aim on the board. We can see how this goal will help them reach later goals and where it falls in the curriculum map.

Our students don’t necessarily see these connections.

When I visit classes, I ask students what they are learning for the day. They tell me, “page 70”  or “the problem on the board.” Sometimes they point to the aim that is written in their book but often it’s not in kid-friendly language so they don’t know what it means when I ask them about it.

It can be hard for a child to know what they are supposed to be focused on. Sometimes we are lucky if they even know what they are supposed to be doing in class–never mind what the goal of it is.

That’s why clear learning targets are so important.  Think about these questions:

  • What do you want students to walk out of your room knowing that they didn’t know before they walked in?
  • Can students explain the goal for the day in their own words?  Do they know how it will help them achieve larger goals?  Do they see the connection to pervious goals?

I’ve seen changes in achievement when students are clear on what their goals are.  When students take ownership of their learning and their progress (and I’ve seen this happen even in early grades), they know what they should focus on during class, they know when they are successful and they know when they still need to work on something.

The research supports my experience. Students who can identify what they are learning significantly outperform those who cannot.  Educational researcher Robert Marzano reviewed the research on goal setting and found that student achievement increased 16 to 41 percentile points when students could identify what they were learning.

In my experience, it also helps students change their attitude towards math. Just like running 13.1 miles seems impossible to me, so do goals like passing math class or getting a high grade on a unit test or project for some students.  Breaking down what it means to get there helps students see exactly what they need to do. It also helps them see progress so that getting there becomes possible.

Want to know more? Check out the following:

Marzano, R. J. (2006). Classroom assessment and grading that work. ASCD.

Moss, C. M., & Brookhart, S. M. (2012). Learning Targets: Helping Students Aim for Understanding in Today’s Lesson. ASCD.


The Power of Frequent Feedback


Happy New Year All! I hope this year brings you lots of laughter, learning and new adventures.

My big adventure for the new year is to run a half-marathon. I started running in the fall and in that short time, I’ve learned a lot about myself. One thing is that I respond really well to frequent feedback on how I’m doing. I like knowing how much further or faster I ran than the day before.

A little into my running, I downloaded an app that tracks my mileage, elevation, and time as I run. My favorite part is that is sends me emails when I’ve hit my personal best in various categories.

I love finding out that this week was my personal best in miles ran or that today I ran my fastest mile. What I also love is that when I finished my first 10K in December it didn’t matter that I was one of the last people to finish (notice in the picture how no one else is around me) because it was a new personal best for me.

This got me thinking about the type of feedback we give our students in math class. As a student, I only remember receiving feedback on summative assessments–end of unit tests or when I got my report card. Even then, it was only a number or letter grade. I never received daily or weekly or even monthly updates about whether I hit a new personal best.

How different might things be if students received feedback about their personal best in math?

The research supports the idea that frequent feedback is important for our students. One study showed that over a school year, the rate of learning in classrooms that used short cycle (within and between lessons) and medium cycle (within and between units) assessments was about double other classes.

There is something to be said for giving students frequent feedback. However, simply testing students more frequently is NOT the answer.

Formative assessment is complicated. It involves many things, including thinking carefully about:

  • How students are assessed
  • The type of feedback given
  • How students can be encouraged to take ownership of their learning
  •  What adjustments need to be made to instruction

I’ll share some ideas from the research on how to begin to think about these aspects in future posts.  In the meantime, I’m going to keep trying to reach a new personal best on my runs.

Want to know more?  Read: Wiliam, D., Lee, C., Harrison, C., & Black, P. (2004). Teachers developing assessment for learning: Impact on student achievement. Assessment in Education, 11(1), 49-65.