Intervention: reteaching, review, and problem solving

In my role as interventionist I’m writing the middle school math curriculum for our Title 1 after school program. Circumstances outside of my control have prevented us from rolling out the program sooner, but invitations will be going out next week and we anticipate a start date within the next two weeks.

I know students will need review and reteaching of concepts, but I also want them to develop a repertoire of problem solving strategies so I will be including James Tanton’s Curriculum Inspirations. which is supported by the Mathematical Association of America. It’s a fabulous resource that includes a series of 10 strategies with examples.

Since the first lesson will focus on multiples and factors students will be introduced to Tanton’s think aloud on the Divisible by 13 task using the problem solving strategy Engage in Successful Flailing. They’ll solve the problem using the attached.


Of course students will have opportunities to practice LCM. Then they will have a chance to explore multiples and 3 consecutive numbers problems from Don Steward’s site.

consecutive numbers

Several teachers will be implementing the after school program, so it makes sense for the curriculum to be delivered using Canvas. Each lesson begins with a content page that identifies the learning objectives and mathematical practices. The page also includes a resource link so teachers have access to answer keys stored in google drive.


As the title of this post suggests I want to balance review, reteaching, and problem solving.  Students need to be exposed to challenging problems but they need to have access to them as well.


Nutting things out and intellectual play

This week’s #EduRead is the topic Constructive Struggle from the article Faster Isn’t Smarter. Author Cathy Seeley states students should be challenged with more complex problems. I agree. End of post.

What spin can I offer on this article? For one, I wish James Tanton was Cathy Seeley’s editor. He’d ditch the title and rename it, “Nutting Things Out.” His word choice speaks volumes about his approach to math. While I imagine the two would both vote for more math complexity, the Seeley read is a tad highbrow and offers no concrete suggestions on infusing more complexity. I’m a fan of Tanton. His playfulness and enthusiasm are infectious. Watch one of his videos and you’ll get a google of ideas on nutting your way through things…


as well as engaging in intellectual play.


In addition to the problems posed in Tanton’s videos and books, Dan Meyer and Michael Pershan recently discussed less scaffolding as a means to make problems more complex. Yet another alternative is Marian Small’s book More Good Questions: Great Ways to Differentiate Secondary Mathematics Instruction which offers Open Questions and Parallel Tasks.

Suggestion 1: Open Questions

A number pattern includes both -3 and -13  as terms. What might be the general term of the pattern?

Suggestion 2: Parallel Tasks–students are presented a task and can choose one of two options

Can  -3087 be in the pattern described by the given pattern rule? How do you know?

  • Option 1: The pattern rule is: Start at 9. Keep subtracting 3.
  • Option 2: The term value is 4 times the term number +3

So now you have more resources for Constructive Struggle. I mean nutting things out and intellectual play.

Integer choice assessment part 2

The other day I posted a work in progress–an assessment menu for adding and subtracting integers. Matt Coaty’s comment inspired me to explore how students would share their understanding, as well as develop assessment criteria.

Today I’m unveiling a new and improved version. I’ve taken the idea of a literary book club meeting where students bring a discussion tool to talk about a novel and have turned it into a “Mathematics Symposium” where students bring activities they’ve created about a concept and share their mathematical understandings with the group.

Here’s a revised 2-5-8 menu with descriptions of the criteria for success, suggested product ideas, and a check on whether students have met the expectations, or if they are not yet there.


Each student will have up to ten minutes to share their activities. When the group is finished they will assess each other using this two sided rubric/checklist. If a student marks any area as superlative (italics) they are to indicate on the reverse side what distinguished it from the other choices.

symposium reflection

symposium notes

To determine a “final” grade, I’ll review the artifacts along with the students’ input. I’ve successfully used this format when I taught literature and it was quite successful. I’m looking forward to seeing how this translates in a math class.


Standards based choice assessment

For the past couple of years our math curriculum committee has been focusing on creating performance based assessments with leveled problems. Now that we have a clear vision of the standards and what we want our students to know and be able to do I want to start offering some variety, some choice.

I’m toying with the idea of offering choice once per quarter. Here’s a 2-5-8 menu I created for adding and subtracting integers.

integer menu1

integer menu2integer menu3

In addition to the differentiated content, I like how this format requires the student to do much more thinking and problem solving because they have to come up with their own scenarios and problems. Even the Knowledge and Comprehension levels ask students to think.

One element that is missing is differentiating by product. What options do you suggest? Ideally they shouldn’t take hours to assess. Also if you have other activities to suggest or other input please comment.

Introducing systems of equations: graphing

Many of my pre-algebra kids continually seek context, so I thought I’d head them off at the pass by introducing graphing systems of equations as an exploratory activity using a couple of word problems. Here’s one of them:

systems problem


We’ve been graphing linear equations for a while so I purposefully added the “How could graphing each equation help you?” question to guide their thinking.

I came to learn that some students would rather not have the context and just be given the systems to graph–it’s a lot easier. “I hate word problems,” said several students.

That’s my fault. I need to do a better job of applying these skills in context. After we talked about context it was time for some graphing practice. As I checked their work I was reminded that a few haven’t improved their “number sense” when it comes to graphing. I wish I had snapped a photo of one student’s work, but I’ve recreated it below.


My questions to her were: 1) While your table is accurate, why did you choose values that fall outside the graph?  2) How could the slope and graphing the y intercept help you?

This particular student doesn’t recognize how less accurate her lines will be if she picks points beyond the coordinate plane that’s provided. If students are graphing using a table, it’s a great opportunity to discuss when it’s appropriate to choose negative values in order to stay within the graph.

In previous conversations with this student I also know she doesn’t have a strong grasp of the meaning of rate of change. I need to get her into math lab.





Students learn to give feedback with a growth mindset

Yesterday Algebra’s Friend introduced me to a new professional development blog Read…Chat…Reflect…Learn! While I enjoy reading blogs for lesson ideas, I also love reading journal and magazine articles, books, research studies, etc. as another way to stay current. I’m glad I found this blog and I’m thrilled to see that it’s in its infancy, only because it makes me feel that I haven’t missed out on too much!

The current topic is feedback. The article for discussion was How Am I Doing? It offered a good overview, but what was missing was how to provide feedback using a growth mindset. This excerpt, Types of Feedback and Their Purposes,  gives clear examples. One type of feedback I tend to focus on is descriptive as opposed to judgmental feedback.

When examining student work for feedback I’ve collected their work in progress, and provided descriptive feedback. I won’t kid you. It’s time consuming. But an idea was floating in my head. What if I taught students to give each other feedback with a growth mindset?  I don’t want you to think I am shirking my responsibilities, but could math students offer feedback similar to students who participate in peer editing compositions?

After reading Algebra’s Friend post I thought I would give it a try. Yesterday and today I showed my students this Would You Rather problem. They were to work in groups of four to solve.

After the groups worked for about 10-15 minutes, I collected their work, redistributed them to different groups and asked the new group to provide feedback. Here’s an example:


Kids being kids, their feedback was somewhat judgmental and not of a growth mindset. Had they rephrased their wording to questions such as, “How could labeling help explain your thinking?” or a statement such as “The final answer is not clear,” would definitely put them on the track towards feedback with a growth mindset.

Doing this activity in groups made it quite manageable as only six responses were being critiqued. While I monitored each group I asked them what questions they had about the other group’s work. I also asked them to be positive with their feedback. Each problem was circulated twice. Doing so also gave groups the opportunity to see how others approached the problem and perhaps revamp their thinking.

Here’s how the above group implemented the feedback:



The next time I see them we’ll continue the conversation of how to offer feedback with a growth mindset. With more practice they’ll get better at it.

I should have thought of this at the start of the year.




Lost in the halls: distance between two points task

It’s been a great week, and a rough week. The highlight was having my pre-algebra students experience the Distance Between Two Points task.  I discovered it among a collection of tasks at the Complex Instruction Consortium website. I tweaked it a bit, and what I really liked about the task was that it got the students out of the classroom. They liked that too. In fact at some point I lost “visual contact” with every student but they never took advantage of the situation!

Students were anxious to get started, so many didn’t pay attention to the directions we read aloud.

distance task directions

Student: “I don’t get it!”

“Read the Resource Fact.”

“So we have to count the tiles?”

“Is that what the resource fact is suggesting?”

distance count

Student: “But how do we find AC and BD?”

To give you an idea of the layout of our building here’s what the kids were doing. I did NOT give them this diagram. They had to come up with it on their own, which was extremely helpful when describing the group’s procedure.


Here’s an example of the student work. They’re getting better at constructing an argument and explaining their reasoning.

distance student work

Letting them be kids with the Pythagorean Theorem

My slightly modified version of this Teaching Channel lesson on the Pythagorean Theorem started a bit slow but within 10 minutes it gained incredible momentum. By the end of the lesson students were set free to search the hallway for problems and gather anagram clues to solve a Who Done It.

Kids were in partners staring at this image from the SMARTboard file I created for the lesson. The rulers followed the directions on the left. The non-rulers followed the directions on the right.

pythag instructions

At first the kids didn’t want to play with the numbers. I wasn’t about to lose at this waiting game so I announced, “Let’s push the pencil and those calculator keys.” One pair began working, then another, then another. Soon the entire class was engaged.

A few minutes went by and one boy asked, “Can the numbers be used more than once?”

“Great question! Yes!”

We had a great discussion on the operations needed to get 10. Not one mention of a formula.

I then asked, “How could we apply what we just learned to golf. How far would I need to hit my tee shot if I wanted to get a hole in one?”

dogleg left

It’s at this point where I tweaked the lesson to my personality. Sidebar: I love the Muppets!

pythag kermit

what's different

Here is where the knowing the vocab came in handy. After pointing out the legs and hypotenuse, students then realized they would need to do something different to solve Question 3.

They practiced solving these:

right angle problems

Then it was time to set them free. The students got this handout.

pythag worksheet

The first two problems were done in class, but the other three created a mini scavenger hunt in the hallway. After correctly solving problems #3-5, each pair received a Who Done It clue. In the Teaching Channel lesson students received their clues in a text message. I chose human interaction because I prefer it.

Here are the anagram clues.

Mustard, kitchen, dagger.

The lesson continued the next day exploring right, acute and obtuse triangles using perfect square cutouts from AIMS Education.

Trick photography asks: “What size is the missing wrench?”


Technically the image isn’t trick photography but the gap in positioning makes you think the size of the missing wrench must be between \frac{1}{2} and \frac{9}{16} of an inch. It may be a form of trickery but could a student argue the missing wrench size is  \frac{17}{32}? How would they justify the break in the pattern?

wrench fractions

One of the wrenches is indeed missing. Would students think to mentally close the gap and consider the possibility that the first or last wrench in the sequence is missing? What would be those sizes?

The idea is to let students explore common denominators and the patterns they discover in this task.

I’ll let you know how it goes.

I’ve never been out in the hall before! I feel so SMART!

My SMART goal to raise self efficacy was to implement four tasks in the first quarter. I only had two. My standard students experienced the Money Munchers and Variety Show tasks. My pre-algebra class had Molly’s Locker Combo task and the Boomerang task. While I didn’t reach my goal, I am making a wee bit of progress with one student’s self efficacy. I have to admit, it’s not me. Her teacher last year had her in math lab nearly every day.

On Wednesday, she was elated that she was part of a group that was allowed to work in the hall on an extension activity while I retaught a concept. It was the first time she ever experienced such a thing. “I’ve never been out in the hall before! I feel so smart!” Since the start of the year she’s only had to reassess one time. Attending math lab has paid off.

The extra time we devote to our students may or may not pay off this year, but it will in the future.