Tag Archives: math

Grammar of Math

I am getting into a regular routine of using VNPS (Vertical Non Porous Surfaces or working at “boards” as I call it with students) and am noticing a couple of things. The philosophy states that students should either be doing open ended problems or “thin slicing”–doing progressively difficult problems that challenge students to discover ways to solve math based on their existing background knowledge and number sense.

Much of what I teach in 6th grade is an introduction to basic concepts that are the foundations of algebra: order of operations, distributive property, exponents, and one step equations. Some of these things cannot be “discovered” through inquiry by students but must be taught as definitions. I call them the “grammar” of math. Rules for the order of operations are a good example. One can demonstrate the need for them with different solutions, but the agreed upon rules must be explicitly taught.

Recently I used whiteboards for progressively harder problems writing and solving exponents. I noticed is two things that break the “rules” of thinking routines. First students needed me to tell them if it was correct or keep working. They have no way to double check their work to prove if they are correct other than comparing to other groups who may also be wrong. Second, I made laps around my room either telling students to retry or giving them a new problem. As we progressed, I could not keep up. Students started getting bored and off task a bit. Nothing major, but playing with markers or erasers and conversing with other groups.

I don’t see this behavior when I give an open ended task because students are more invested and stick with it. What I now would say is that my exponents was not really thin-slicing at all but practice problems (the book calls them Check Your Understanding or CYU) done in a group at the boards.

Sometimes we have to explicitly teach a specific skill or mathematical technique. For example my curriculum requires students to use factor trees for prime factorization. Now factor trees are not really a new technique, but one they learned in previous years. What is new is writing the prime factorization at the end–which again is a grammar thing as it has to be written a certain way.

Variety of Approaches

So here is the approach that I am developing about when to use or not use whiteboards. I will use the introduction of exponents as an example.

  • Day 1 Introduce Concept: Students work on an open ended problem at the whiteboards without vocabulary. We used the tax collector for exponents and followed it up the next day by reading One Grain of Rice together.
  • Day 2 Introduce Vocabulary and Group CYU: I explicitly teach vocabulary and techniques based off from previous day’s experience and then students practice at their boards (group CYU). I taught them the vocabulary for exponents and showed them how to write and solve them. Then students solved progressively more difficult exponents at the boards.
  • Day 3 Individual CYU and Assessment: Students practice problems from the book at their seats. Answer keys are posted so they can freely check their answers at any time. They work with a neighbor as needed, and I roam the room helping. We finish with a quick assessment so I know who has understanding or needs reteaching.

The days are not rigid as I sometimes mix several of these things in the same day. But what I am realizing is that my Day 2 is not really in line with the Thinking Classroom, but it is needed. It is a form of “we do” but without much help from me. I think my students benefit greatly from group practice.

In reading some posts and comments in the Building Classrooms Facebook group, I am not alone in seeing that the thinking classroom approach is not necessarily all or nothing. My Day 2 actually does require the whiteboards at all but could be a game (traditional or online like Gimkit), stations, card sort, partner work, or any other kind of hands on practice. I have been using whiteboards primarily because my 6th graders like them and I don’t have time to create hands on activities every day. As a goal I will develop one strategy such as stations per unit and slowly build things over time.

Learn with me!

If you are interested in how your school can use a PBL framework to teach SEL skills. I would love to have a conversation on how I can help. I have limited availability for PBL & SEL workshops during the school year so contact me early. Check out my workshop page or drop me an email at mikejkaechele@gmail.com. I would love to chat and co-plan meaningful PD for the educators at your school.

Pulse of PBL

The Magic of Vertical Whiteboards

Students working on 1001 coins via @ankermath

As mentioned previously, this year I am committed to getting students to think, rather than just mimic me in math class. A huge component of this is working in small groups at vertical whiteboards (VNPS-doesn’t have to be whiteboards) around my room. Each day students solve either open-ended problems or practice on progressively more difficult problems at their boards developing their own strategies and building on their background knowledge. So far it has made a huge impact in my class.

While vertical whiteboard use is very popular in math classrooms, it is a flexible strategy that can be used in any content area. In science, students could problem solve or document experimental data that they turn into graphs or charts. In social studies students might create timelines or outline key events. In English, it is a great strategy for collaborative mindmapping, brainstorming, or outlining a draft of an essay.

Vertical whiteboards are engaging and a great pedagogical practice to add to your tool bag. Here’s four highlights of why you should try vertical whiteboards too.

Via https://www.globemedical.com.au/adelaide/interact/blog/sitting-is-the-new-smoking.html


Kids sit too much in school. Their bodies need movement. It is not a luxury but a biological need for both their muscles and their brain. I think that most educators realize this, and some teachers use brain breaks to introduce movement into static lessons. But oftentimes adding movement feels like another thing to include in an already overcrowded lesson plan. On the other hand, consistent vertical whiteboard work creates a culture where movement is a part of every day routines.

Whiteboards allows for age appropriate movement. Students don’t need fidget toys because they can move freely in their area unrestricted by desks or the need to “stay still.” I played some quiet music one day this week and noticed several students “dancing” unconsciously while working on the math. Movement while working at boards is natural and individualized as each student moves as their body needs too.

Collaborative Thinking

Students attack the problems together practicing teamwork. There is plenty of research that shows students learning from each other is an effective strategy. Students are able to combine their background knowledge to attempt a solution together. Since students can erase their work and start over at any time, they are free to make mistakes without judgment.

Another key aspect is not bailing kids out. When I restrain my natural tendency to over help students-they think. So instead of answering questions, I respond with questions to point them in what they need to analyze.

This past week, students were struggling with one part of a story problem. Students would ask me, “we need to divide, right?” Instead of bailing them out, I explained that it could be solved with addition, subtraction, multiplication, or division. Different groups tried different approaches. After a long struggle when most groups had solved it, one group finally figured it out. A girl in the group jumped up and down saying “we are so smart” after sticking with it and finding a solution. I am seeing students confidence in math grow daily.

Shift in Control

Previously I talked too much in class. We spent a ton of time following the textbook notes in the format of “I do; we do; you do.” At the end, I noticed two things demonstrating how ineffective the notes were. First, we often only had 5-10 minutes left for the “you do” part. Too much time was spent on “I do.” Secondly, I still had many students who needed my help and could not solve problems on their own even though I had just demonstrated it several times and they had examples written down. Clearly it wasn’t working for them.

At the boards, students are working on math and talking about math with each other. They are negotiating strategies and approaches. They are checking each others’ work for errors. Instead of following a prescriptive algorithm that they don’t understand they are using and building their own number sense. It fits my manifesto: “whoever is doing is learning.” I have turned the time of class over to students to work on math instead of aimlessly listening to me.


My sixth graders naturally love working at the boards. After forming random groups for the class period, they immediately get to work solving problems. I have experienced zero resistance or complaints to working on the boards. Students are disappointed when I say that we are done and are shifting to something else. Previously, I heard lots of complaints and resistance to taking notes or practice problems.

Many students say, “I don’t like math, but I like this class.”

Will kids get tired of the boards? Is it just a novelty? Time will tell. So far I have mixed in a game, Prime Climb and the Exploding Dots interactive to work on number sense. If needed I can “spice up” board work, but my guess is that students won’t get tired of boards if the tasks are compelling.

I am enjoying class and having fun too. It makes me more positive and light-hearted. I am less stressed. Kids are loving math class and telling me daily.

Learn with me!

If you are interested in how your school can use a PBL framework to teach SEL skills. I would love to have a conversation on how I can help. I have limited availability for PBL & SEL workshops during the school year so contact me early. Check out my workshop page or drop me an email at mikejkaechele@gmail.com. I would love to chat and co-plan meaningful PD for the educators at your school.

Pulse of PBL