Book Review: Accessible Mathematics: 10 Instructional Shifts That Raise Student Achievement by Steven Leinwand

For our first book review in MTH 495 I chose to read Accessible Mathematics: 10 Instructional Shifts That Raise Student Achievement by Steven Leinwand. The book was a wonderful short read that provided lots of information regarding teaching mathematics. Each instructional shift is an easy adjustment to make within an already established classroom, or a good foundation to start a classroom.

Most of the instructional shifts I have learned about at Grand Valley State University, so it was really beneficial to reiterate important ideas about the classroom. Leinwand did a great job of providing examples of dialogue within a classroom with and without the suggested shift. This provides a good idea of what our students are not getting out of math class and why so many students are bored with math.

Chapter 2 discussed the importance of constant review. This is one topic we did not discuss in my education classes and I found to be interesting and something I want to bring with me to my future classroom. It discusses the waste of the first few minutes of class when the teacher takes attendance, checks for homework completion, and then attempts to get class going. A simple solution and easy way to take full advantage of the short class periods is to have mini-math “quizzes” to get students minds warmed up for the rest of class. There could be 1 to 7 problems that cover topics from the first week of school or even foreshadow next weeks lesson. This provides students with a chance to master the variety of math skills. Let’s face it, its rare for a student to master something after half of a week of instruction. Plus, the teacher is able to check homework and do attendance while the students are doing this.

I specifically see my self using the idea of constant review in my future class room but adjusting it to fit the needs of the class I see my self having and to most importantly fit my students needs. I see my future class as being a very much activity based class which asks for as much time as possible for students to work on these activities. Therefore, I think I would stay away from any number of questions that exceeds 3 or 4. I believe it would be very beneficial to use the five minutes that it takes for me to take attendance as a time to have 3 questions for my students to work on. One of these questions I would pull from previously learned concepts earlier in the year. This as described above will allow for mastery of the information rather than memorizing it for the test and immediately forgetting it. The second question I would think would come from the current unit being taught to keep it fresh in my students minds. Possibly directly from the lesson previous to that days lesson just to reiterate what we learned the day prior. The third question I foresee coming from the future unit. Students I would expect to struggle and possibly have no clue how to work on this problem and I think this would be great for the class. I would anticipate the discussion of the first problem to go fairly quick because hopefully they would have it mastered. The second problem I see spending more time on simply because we would of just discussed it the day prior. The third we may not even come to a final conclusion. I see it being more meant to get the students thinking ahead so that when it comes time to actually teach that concept, students are already aware of it which I think will make the learning experience much better.

An important instructional shift Leinwand brings to our attention is having real world connections and situations for students to apply their math skills in. This will engage the students in their learning instead of falling asleep while counting down the minutes to go to phys. ed. class. With having these real world situations students can discover math and have in depth conversations amongst their peers and really learn the material. I see my self using a lot of videos to bring a real world situation into the classroom. This will get my students hooked onto that video, and then I can supply activities that feed off the video shown.

I see my classroom being a very activity and discussion based. After class activities, I would like to have class discussions about the activity where I can ask students questions such as “why?” or “how do you know?” In my experience as an educator thus far, I have found when I do this type of thing students learn a lot more. One, because students have to develop enough knowledge to be able to explain their thought process. And, secondly the students who are having trouble have multiple opportunities to hear explanations of the topic we are covering. This type of environment is tightly related to Leinwands 10th instruction shift of making “Why?” “How do you know?” and “Can you explain?” apart of your classroom mantras.

Another shift Leinwand talks about is having multiple representations. This is something I have near the top of my list as far as how I would like to teach my students. There are so many different learning styles out there and a big reason so many students don’t like math is because there teacher only stands in front of the class spewing out information that the students have no clue what it means. I am going to make it a priority to constantly give my students opportunities to draw and describe their math. The board will be a log of the class period full of pictures, tables, and number lines to give my students a visual to go off of to hopefully help them make sense of the information.

This book did a fantastic job of pointing out important shifts teacher need to make to better the learning of their students.  would recommend this book to any one entering the field of teaching or even current teachers. These are 10 minor adjustments that can be made in a classroom that can make major impacts for the betterment of our students. Leinwand does a phenomenal job of portraying the importance of each.




Math: Fibonacci

During last weeks class we watched a YouTube video called “Doodling in Math: Spirals, Fibonacci, and being a Plant part 1 by ViHart. The video talked about how Fibonacci numbers are everywhere in nature, specifically items that started with “pine” and flowers (From the video.) There spirals within these plants that correspond to Fibonacci numbers. Such as 5 and 8 on a pine cone, or 21 on a sun flower. One part of the video that intrigued me was when ViHart created a pine cone using spirals.

I am not an artsy person to say the least. Thus, it was really cool to see some art being made using mathematics such as Fibonacci numbers. So I decided to make a pine cone exactly like ViHarts. With 5 spirals going one way, and 8 spirals going the other. Yes, I agree this isn’t very creative, at least I should have used two different Fibonacci numbers but I am trying to learn how to be a decent artist, so I was using the visual aide to help me. I struggled to create a decent looking piece.

Some of my issues included:

  • Making my spirals with the right amount of space between them
  • Knowing where to make the “petal” of pine cone and how big to make it

The steps of how to create a pine cone include:

  1. Pick two Fibonacci numbers (I chose 5 and 8)
  2. Make 5 spirals going one way, and 8 spirals going the other way
  3. Create your pine cone by going 5 one way, and 8 the other using your spirals as a guide (This was very tricky for me)

Below is a picture of the best outcome I was able to come up with.


As you can see, even with the help of Fibonacci, I am not the best artist. In the middle of the pine cone, you can count and see that I am following the Fibonacci numbers 5 and 8. Although, as I get wider, I start to lose where I should be drawing the petals and count nine of them on the outside instead of 8. I would say this is a start into the right direction of creating a decent looking pine cone. I am going to continue to draw these using a variety of Fibonacci numbers. I will be posting the results on here. Maybe I will have to resort to Geogebra to get something that looks good. Even if I did enjoy drawing free hand using mathematics that I have learned.

Doing Math: Tesselations

During our last class meeting we “did math” by creating Tessellations. Below are a couple that I have created using Geogebra.

When I first started to create these tessellations my first thought was to start out with a single shape then build off of that shape. For the first one, I decided it would be easiest if I can use different figures off of my initial shape, the octagon, so that the end result would be a square with multiple shapes within. I then started to play with the reflection over a line tool in Geogebra and started to make my tessellation bigger along with having symmetry. As I started to make the tessellation bigger into a bigger rectangle I thought it looked boring. Therefore, I decided to bring the octagon from the start all the way to the finish by making the shape of the final tessellation exactly that. An octagon. Well almost an octagon, the 4 corner sides are not equal to the other sides. But, either way I think it brought some more spunk to my tessellation.

For my second tessellation, I wanted it to be much more creative. I wouldn’t consider my self the most creative person when it comes to artsy type tasks but I was happy with the outcome. I started with the same mentality as the first by making a single shape my starting point. I used the triangle, which eventually turned into a hexagon. I went to the reflection over a line tool and just started to play with the different directions of reflection. I  admit, I accidentally came up with the shape of the upper half just my reflecting and thought it looked cool, especially have I reflected it over a line to create the bottom half. I then referred to some tessellations we looked at in class and noticed how colorful they were. I wanted the symmetry to still be there even within the color selections, so I made sure to put color in the same spot on the other side of the symmetrical line as if I was painting the wings of a butterfly.