Wednesday, May 31, 2017


White Male, 3D Model, Isolated, 3DWhen I went through my teacher training, it was at a time when you were told to assign a bunch of problems so students gained more practice.  Since then research has emerged indicating homework is not that effective.

One training session I took indicated you should assign homework but not many problems and you should include the answer so students know where they are heading.

I also tend to choose problems covering material students have already had time to master.  I seldom assign homework based on what they learned that day because most students need time to master the material.

Its also said that students need immediate feedback but when correcting homework, it usually takes the teacher time to check how the student got to the answer.  I seldom check only the answer because I gave it to them.  I've found students will write something down and then put the correct answer down.

 I've also heard the teacher should show all the work on an answer sheet so students can copy it down or check their work if they want to.  Either way they get the correct answer but I'm not sure that is the most effective way.

I just read something that makes sense.  For homework give "suggested problems" but rather than checking to see if the work was done, the teacher gave a homework quiz with only one or two problems from the suggested problems.

To help students before the quiz, it is suggested the teacher answer questions related to the suggested homework problems to clarify understanding.  Then give the quiz.    It has also been suggested the quiz be composed of questions similar to the homework rather than the actual homework questions themselves.  In the same article, the teacher used a rubric for the quiz rather than actually grading it.

Recently, people have developed apps which solve problems step by step.  Students can use these apps to solve the problems without ever learning the material.  To counter this perhaps some of the homework problems should require students to explain what happened at each step.  Or maybe there should be a combination where students solve some problems while explaining other problems.

I'd love to hear your thoughts on the topic of homework.  Please let me know what you think.

Tuesday, May 30, 2017

Misconceptions About Fractions

Fraction, Variables, Math, Division  I teach high school students who still add fractions straight across without bothering with common denominators. I honestly don't know who they get so far without having the basics down.

Several years ago, I worked with a young lady who did not realize when dividing a pizza into pieces, each piece had to be the exact same size.  She didn't know that.

One common misconception is when adding or subtracting fractions you see students pull the 1/2 + 2/5 = 3/7.  As stated earlier, I have quite a few high school students who do this and have no idea why they are marked as incorrect.

Another misconception I've seen is students not understanding when creating fractions with common denominator, they are trying to multiply by a fraction equal to one such as 1/2 x 2/2 = 2/4 so they can have 2/4 + 1/4 = 3/4 instead of 1/2 + 1/4.  Many of my students forget to multiply the numerator by the same number as they multiplied the denominator. Students have been taught they multiply both numerator and denominator by the same number rather than understanding they are multiplying the fraction by an equivalent form of 1.

This misconception comes from teach a process in elementary rather than teaching the concept behind it.  When students do not have this understanding it is difficult for students to find common denominators when working with algebraic fractions.  Although the process is the same, they do not see the connection.

Another thing I've observed is students try to find common denominators when they multiply or divide fractions.  They apply the common denominator rule to all fractions regardless of operation.  I suspect its because they have not figured out the differences in what each operation represents.  In addition, when dividing they often flip the incorrect term because they are told to flip one rather than understanding they are multiplying by a reciprocal.

Furthermore, students see fractions as always being part of a single object such as 1/6 th of a pizza being one piece out of six but 1/6th could represent one red ball out of 6 balls.   They see the whole as being a single object rather than possibly representing a total number of objects.

It seems some students see dividing a whole number by a half the same as dividing in half.  An example would be 4 /(1/2) is the same as 4/2 rather than 4 x 2.  Its why its difficult to create pictorial representations showing 4 divided by 1/2.  Even I struggle with that one because I never learned the concept, only the process.

I'd love to hear what you think.  Have a good day.

Friday, May 26, 2017


Question, Mark, Question Mark, Surprise  When I went through teacher training, we were taught to ask the student "What is the next step?"  since solving equations was a process.  As we know, things have changed so our method of questioning should be changing but if you are like me, its not.

What are the best types of questions we should be asking our students now.  There are 8 recommendations of types  questions for the classroom.

1.  Use fewer information gathering type questions such as "What is the area formula for a trapazoid."  These have a time and place but do not require higher level thinking.

2. Ask questions which require students to explain, elaborate, or clarify their thinking.  These are probing questions needed to uncover student understanding.

3. Allow students time to answer.  Students need at least 10 seconds to gather and formulate their answers and ELL students require even more time. Research indicates teachers usually allow fewer than 5 seconds to answer.

4. Ask students to make their thinking visible by connecting mathematical ideas with relationships such as connecting arrays with multiplication.

5. Take time to encourage students to reflect and justify their answers.  This forces students to stop and explain their thinking process so as to understand the material better.

6. Avoid guiding the conversation to a predesignated  conclusion.  When you do this, you ignore students who may need to go elsewhere to get to the conclusion.

7. Try focusing your questions on the needs of the students.  These questions will probe, assess, and it encourages students to express their thinking.

8. Encourage students to ask questions of each other. You will have to help students learn the types of questions which are better asked by them rather than "Did you get the answer?" "No?"  "Time for the teacher." 

These are suggested by the National Council of Teachers of Mathematics but I stumbled across a paper in which the person suggested writing questions which were non-computational but could spark conversation so the teacher could gauge their back ground knowledge and understanding.  The teacher sets up the situation with four possible answers.  Each of the answers connects with the situation but is not the whole answer.

Imagine that you are sky-diving. The graph of your speed as a function of time, from
the time you jumped out of the plane to the time you achieve terminal velocity is most likely
a)Increasing concave down.
b)Decreasing concave down.
c)A straight line with positive slope.
d)Increasing concave up.
Notice there are no numbers, just concepts and situations where students have to justify their answer.

Let me know what you think.  I'm interested to hear. Have a good day.

Thursday, May 25, 2017

Lego Apps

Error, Not Found, 404, Lego, Mistake, 4  The other day, I went to the app store for my ipad, looking for lego apps that might be used for teaching math.  I found three but none of them proved to be worth it.

Math 4 Kidz 3 is advertised as using legos to teach fractions but it comes without instructions and so I made a few guesses and never figured it out.  I hoped this might be work so I wouldn't have to worry about losing lego blocks but as stated earlier, it just doesn't work.  If I can't figure it out, I am not going to consider using it in class.

I also found two lego education apps, one for the UK and one for DE.  Both are outdated in terms of one states its for 2016 while the other is dated 2015.  I'll start with the one for the UK. I downloaded two previews, one on maths and one for general learning experiences. It has a preview for More to Maths curriculum packs 1-2.

It shows everything but it looks like its actually the book you'd buy or get with the kit so you know how to use them.  Honestly, I didn't see anything I'd really be interested in.  The DE education one has two issues but they looked more like catalogues and were not particularly interesting.

I even went so far as to look for ibooks on the topic but didn't find any specifically focused on math.  This leads me to believe the most up to date information on using Legos in the math classroom.  So there you have it.  Look on line for better suggestions and uses of Legos in the math classroom.

Wednesday, May 24, 2017

Legos and Math

Lego, Lego Blocks, Stones, Toys I just discovered you can use Legos to teach math in elementary school or for students who need extra reinforcement.

I've used Lego blocks for building robots, houses, etc but never for math. Tomorrow I'll review three apps which focus on using Lego blocks in education but today I'll look at some general applications.

Lego blocks can be used to show fractions as part, part, whole.  or total is 8 for one block as the whole.  The parts might be two blocks of 4 or one block of two combined with one block of six.  Very visual display.

In addition, Lego blocks can be used to teach students about equivalent fractions which is something students struggle with due to not knowing their multiplication facts.

Arrays which is another way of using these in terms of say square arrays or 2 by 2 or 4 by 4.  According to the author of this article arrays can be used to teach multiplication or division.  She includes a link to a 5 page exploration activity.  What is nice is she shows how to use the blocks so students can work on their multiplication facts.  With a small adjustment these arrays can be taken further to explore factors, area, and common denominators. 

In addition, Lego blocks can be used to explore mean, median, and mode by using stacks to represent numbers.  A visual representation for students who have trouble working with just numbers.

This site has lots of links for using Lego blocks in class. Although most of the links in this list are for elementary math, there are a couple which could easily be used in middle school or high school.  There are links for using them to find area, teach ratios, probability, Pythagorean theory and graphing.  Although the graphing is for first grade, it could be used for bar graphing.

This is a cool idea - to use Lego blocks to reinforce math.  I checked at the local Barnes and Noble book store for Legos but they only have the predetermined kits.  They do not carry the ones where you can just build things out of them.

There are also videos on youtube which show how to do this if you want to see how to do it.  Let me know what you think.  I've got a couple of apps I'll report back on tomorrow. 

Tuesday, May 23, 2017

Explaining The Process

Black-And-White, Blur, Book, Business  As you know there is a move in mathematics to have students provide explanations to accompany the steps used to solve various problems.

Sounds good, doesn't it!  But have you stopped to think about what they should be explaining?  My first impulse is to have them write down the steps such as
1.  x + 2 - 2 = 3 - 2.  Subtract two from both sides.

 This meets the basics of the idea because I have explained what was done but I didn't explain why.  Why am I subtracting two from both sides.  The why is as important as the step itself.  Without the why, you are just following a mechanical process.  When I got my degree, the professors never worried about the why's only if we could get the answers. 

As I've searched for ways to help my students learn, I've learned more of the why's.  I've learned how to provide visualization for many concepts.  Visualizations, I didn't need and couldn't have designed by myself because it wasn't important when I got my degree.

To understand why is to understand more.  To explain the why is to have the ability to explain more about the concepts which means students know it better.  Calculations do not equate to understanding. Unfortunately, its an uphill struggle for me in the high school level because too many of my students had teachers in elementary who knew nothing about math so they just followed the book. 

Many of these teachers were scared of math so to them there was only one way to do the problem and only one form of the answer.  I include the comment about form because I know a young lady who lost points on a test because she wrote the answer as (x+1)(x+1) rather than (x+1)^2.  The teacher neglected to state in simplest form.  He just wrote answer the question.  She went back and argued the point and got the points back.

Her math teacher had started out as an English Teacher who switched to teaching Math because there was only one way to do the work and only one form of the answer.  He did this long before they wanted people to be highly qualified in a subject.  He didn't really care about the what or the why.  He only cared that they did the problems his way.

So let me know what you think about explaining the process.  Do you think the why is as important as the what?  I'd love to hear from you.  Have a good day.

Monday, May 22, 2017


Clock, Pointer, Clock Face, Old  My state is currently rewriting standards and the state test to change the way they look at things.  Rather than focusing on standards, they are looking at achievement levels.

I know someone on the committee for math and he's shard a bit of information which makes me a bit concerned.

According to him, one change is looking at multiplication using the standard algorithm.  He said the standard algorithm is for the two numbers lined up, one directly above the other but many teachers are choosing to teach the lattice method because students catch on to that faster.

The long term sub in the 5th grade commented how so many of her students struggled with the two digit by two digit multiplication.  I offered to come in to teach the lattice method.  After 20 minutes, her students had the basics.  The more advanced ones were off doing two digit by three digit or even three digit by three digit but they all got it.

I suggested to a friend that since lattice is being taught more and more it could be considered a standard algorithm especially as most tests have students provide the answer but do not have to include the work.  The test doesn't know which method is used, only if the answer if right or wrong.

I show several different ways to multiply binomials and lattice is one of the methods I use because many of my students already know it so its a short hop, skip, and jump to using it in algebra.  They always get it set up properly but mess up because they do not watch for signs.

So if tests only want the answer, why even throw phrases like standard algorithms into the standards.  Why not just use acceptable algorithm instead since in reality there are so many different ways to multiply numbers together.  It seems that even state standards are still being caught up with processes rather than real understanding of the material.

I would argue our testing should move towards being able to explain the steps of the algorithms in addition to using the algorithms to test understanding rather than just the mechanics. Most tests have moved away from hard copy to computer based so we should be able to design tests which allows us to check their understanding of why certain steps are used.

 There are times I feel like we are wagging the tail of the dog rather than letting the dog wag its own tail.  I'd love to hear what your thoughts on this topic.  Have a great day.  

Friday, May 19, 2017

Visible Thinking in Math

Thoughtful, Think, Thinking, Reflection Yesterday, I looked at visible thinking in general but today its time to look at its practical applications in math.  I read about a gallery walk that could be applied in Math.

Students hang their completed problems on the wall.  This presentation would include information on the steps taken to solve the problem.  Each paper is numbered so students names do not appear.  Every student is expected to check out each piece of work and write two things on sticky notes they attach to the paper.  The first is a question about the work and the second consists of a positive observation.  At the end, the author reads all the questions and comments.  This provides immediate feedback.

A good way to introduce a new unit in math is through the see - think - wonder strategy.  The teacher asks students "As you preview the unit, what do you think you'll be studying? What do you think you'll be learning? What do you wonder about this unit?"  When using questions like this, students need to be given a lot of scaffolding to learn to do it.

When starting students on a problem, use the think - puzzle - explore strategy.  Ask the students "What do you think will happen? What are you puzzled about? What do you want to explore to confirm or counter your thinking?"

For vocabulary, place the words on papers which are spread out around the room.  Have students go through with markers and write down what they believe the definition is or make a comment about the word.  This is done silently because the marker does the talking.  This technique can also be used to have students reflect on their learning.  Post a few questions around the room and have students write their understandings, examples or questions down.

For deeper understanding, ask students what made them say that.  It provides clarification of their thinking.   They can also explain what they used to think and what they think now to show how their thinking has changed.

The idea behind having teachers model visible thinking is because many of us are unable to explain why we do this or that.  We've always done it that way because it was the way we were taught.  By taking time to teach students how to make their thinking visible in math, we will increase our understanding and our ability to express our thoughts. 

Let me know what you think.  I'd love to hear.  Have a great weekend.

Thursday, May 18, 2017

Visible Thinking.

Monument, Thinker, Think, Force, Globe  I've been cleaning the house and stumbled across a book on making thinking visible.  Of course I stopped working to check it out as if it were a new toy.

While reading it, I wondered if visible thinking is the same as thinking aloud or are they different enough to warrant a more in depth look.

Lets start by looking at what "Visible Thinking" is.  There are 6 ideas this topic is built upon.
1.  Learning is a consequence of thinking.  It has been found their understanding and memory of material increases with thinking.

2. Good thinking is important because its how a person invests their thinking that makes learning possible.

3. Thinking develops from social endeavors.  Individuals learn from their social interactions with those around them.

4. When you foster thinking, it becomes visible. For the most part, thinking is invisible so we don't always know how we arrived at an answer.  Effective thinkers make their thinking visible by speaking, writing, drawing, or other method.

5.  Classroom culture makes it more inviting to develop visible thinking.  The classroom needs to have  routines and structures which encourage learning.  It should also have language, expectations, time, modeled thinking,  interactions, and opportunities to think.

6. There should be professional development opportunities for teachers to develop their own thinking.

Below are 4 suggested routines to help students learn to make their thinking visible.
1. Headlines - Have students create a headline for a newspaper which summarizes the most important information covered in the lesson.

2. Connect - extend - challenge. This requires the teacher to ask students three questions.  The first is how are the ideas or materials presented connected to what you've already studied or already know?  The second asks what new ideas pushed or extended your thinking in new directions. The final question is which new ideas or material are you still confused by?  What are your questions?

3. See - think - wonder has the student comment on what they see (making observations), what they think about it, and what do they still wonder in regard to the topic.

4. Compass points - Each compass point stands for a different level of understanding.  East means Excited or what excites you about this idea or proposition.  West stands for worrisome or what do you find worrisome about this idea.  North is need to know or what more do you need to know.  Last is south or stance, steps, or suggestions for moving forward.

So it appears visible thinking is in the same area but a bit different. So next is to look at its application to math.  Hope you enjoyed it.  Have a good day.

Wednesday, May 17, 2017

More Thinking Aloud in Math

The Thinker, Bronze, Sculpture, Rodin  Today I"m looking at ways teachers can model thinking aloud in the math classroom.  I know I don't know how to do this because its not something I was trained to do in my teacher training program.

I don't even think about what I'm doing because I've done it so long, I can do it.  I don't know how to do it or how to model it for students.  This is what today is about.  Modeling it for your students so they understand what you mean.

I found this lovely little pdf from a place in Texas which gives some great ideas for modeling including preparing to model and specific ideas for modeling in math.  The 15 page downloadable document begins with steps to prepare to model using a passage from the text book.

It supplies two and a half pages filled with general strategies and specific sample prompts some of which can be applied to math with minor adjustments.  Then it discusses what to do after the think aloud lesson and includes some questions to ask students.

This finishes out with ideas for applying think aloud to specific content areas including math.  I love it shows ways to use think alouds to the textbook.  Furthermore, it includes ideas for scaffolding vocabulary definitions, pronunciation, etc.

This site is focused specifically on math. It suggests the teacher provide clear explanations for solving problems which include your thinking such as how you decided each step.  Students need to know you are interested in how they got the answer rather than focusing on the correct answer only.  Ask questions on their thinking rather than on the process so they have a chance to explain their thoughts.

Another suggestions is to give students strategies and models which include prompts to help them think aloud.  This document provides examples of prompts.  In addition, it is suggested students jot down notes as they use models and diagrams to visualize their thinking.  Encourage students to share their thinking when they get the correct answer, the correct answer with the wrong reasons, or incorrect.  Students learn well from each other so sharing their thinking under all circumstances helps them learn better.

The final suggestion is to provide ongoing formative assessment.  Often this would be having one student explain their thinking to another while the teacher concentrates on the content and carefully chooses the best way to respond based on their knowledge of the student.

Don't worry, I'll be back to revisit this topic as I find more interesting information.  I hope you get some good info out of this.  Have a good day.

Tuesday, May 16, 2017

Dividing Fractions in Real Life

Fraction, Symbol, Icon While writing yesterday's entry, I realized I could not name any believable situations when a person would need to divide fractions.

I started with Dr Math and his examples included one my students would roll their eyes on.  It was about having 3/4 of a pizza left and wanting to know if four people could share it so each person got 2/5th of a pizza.

There is always someone in the class who would say, yes they could all share but not necessarily what they want since its in fourths, not fifths.  Another might say, who cares, they'll eat what they eat.

Another example which is more realistic states you have so much land.  You want to subdivide the land into smaller plots but the county requires that the septic system needs x amount of land.  How many plots can you sell.  This is better but its hard for my students because the houses either use a suction system or a honey bucket ( a bucket for collecting human waste)  They have no idea what a septic system is.

Another site provided some great real life examples such as you buy 7.5 yards of material to make up several pot holders.  Each pot holder requires 3/4 foot, how many pot holders can you make.  Or so many feet of rope that must be divided into lengths of 1/2 foot, how many strands will you have. Or the same using ribbon.  These are actually one's I've done myself when making things for the house.

Dividing fractions occurs in cooking when you want reduce a recipe.  I have tons of recipes which make enough for 6 and there is only me.  This means I either eat or freeze the rest of the meal or I cut the recipe so it makes enough for 2 or 3 people.  My students have such large families, they don't usually try to reduce a recipe.  Cooking around here is just throw stuff in till you have the right amount of food. 

The reality is we can come up with real life situations for dividing fractions but most of those situations are not ones people will do such as figuring out exactly how much detergent is used in each load if you have 50 oz of detergent.  I just scoop and dump.  I don't sit there and calculate the number of ounces used each time.  Besides, most instructions require you to use more than is actually needed.

A few years ago, I taught a remedial math class where I had students draw pictures visualizing dividing fractions.  I am the first to admit that it was hard because I'd never done it.  Fortunately, I had a book which showed a few problems so I could figure it out but it was not easy.

I'd love to hear some real examples which are used by people so I can share them with my students.
Thanks for reading, have a good day.

Monday, May 15, 2017

Connections in Math

Network Cables, Network Connector, Cable  I'm currently reading a book "Writing on the Classroom Wall" by Steve Wyborney which is filled with big ideas that apply to learning.

The first one is "Learning is about Making Connections".  Making connections is one thing my students struggle with daily.  They approach each topic as if it is totally isolated.

I constantly struggle with trying to help them make connections.  Steve includes some great examples, including ones I can use in my lowest level math.  I like the ideas he presents but the review of the book is for another day. 

The first big idea brought up the need for students to connect ideas together in math.  I came across an article from the National Council of Mathematics discussing this very topic.  The example the author provided gave me an "aha" moment because I've never seen the connection.  At some point in elementary, middle or high school, teachers cover multiplication, division, multiples and factors.  When the class used manipulatives to find the area of various polygons, a student pointed out this was just like using factors and multiples from earlier in the year.  Give it some thought and that student is absolutely correct.

Too many times, both students and teachers rely on tricks or procedures without going any deeper?  My students cannot explain why when dividing fractions, they have to flip the bottom fraction and multiply.  I've covered it in high school math because my students have never seen the explanation.

The NCTM states all students should:
1. Recognize and use connections among mathematical ideas.
2. Understand how mathematical ideas connect and build upon one another to build a coherent whole.
3. Recognize and apply mathematics in situations outside of mathematics.

This article actually reinforces the idea of showing all the steps even if the students are not thrilled with all the micro steps because they are so used to using shortcuts and tricks. 

This spring, I started looking at reorganizing all my math classes so I group similar processes together to see if I can help students see connectivity.  For instance, review exponents and rules of exponents before going on to rational exponents, logs, natural logs etc which all use the same rules rather than teaching them at various points throughout the course.

I'm thinking of reviewing regular fractions and teaming it up with algebraic fractions since the process is the same for both.  I'm hoping by starting with the simpler review, students will have a chance of extending their understanding that one situation is actually a more complex application of the other.

In the past I've reviewed the lattice method of multiplying whole numbers before showing students how its used to multiply binomials.  The process is the same.  I also have used drawings of the same type to show both the multiplication of numbers and binomials. 

I'm hoping to utilize the concept of Big Ideas next year to show the umbrella idea applies to more than the one thing they've concentrated on the past. 

I'd love to hear what you think about this.  I appreciate when I hear from readers.  Have a great week.

Friday, May 12, 2017

Learning to Read Graphics.

Analysis, Analytics, Business, Chart  While researching a few topics, I stumbled across several article dealing with teaching students to read graphs.  This morning, I realized we should be teaching students to interpret graphs because reading simply means literal interpretations while interpreting indicates you are reading it and associating meaning with the graph itself.

I wondered if most high school teachers assume students arrive knowing how to interpret graphs.  I've discovered most of my students can sort of read most graphs but if asked to extrapolate meaning, they stutter to a complete stop.

Before looking at ways to teach this topic, it is important to look at the reasons students struggle with this topic.

1. Context - It is important to know how and why a certain graph was created.   In other words, what was the question asked which produced the final graph.  Without this information, students often struggle with understanding why its interesting or how it relates.

2. Abstraction - Students and teachers often relate to the material differently due to experience.  Sometimes students do not understand the mechanics used to create the graph.

3. Active vs Passive - Lecture vs doing it themselves.  They are more likely to learn if they are asking questions, working with the data, or discussing it with others.

4. Everyday Tool - Often interpreting graphs is reserved for when students study certain topics rather than seeing it as a tool people use for communication everyday.

It is suggested students be taught to construct the graphs themselves from given data.  This provides a context for the whole graph.  I remember in high school when we looked at pie charts for someone's monthly spending.  I didn't see why I needed to know they spent 30% on rent, 25% on food, etc until I bought my first house.

The loan officer explained our house payment could not be more than 35% of our income or they would not loan us the money.  I finally had a context for this type of graph.  Without it, most students have no reference and that makes it harder for them to relate to it.

Most of the things I've found designed to teach students to interpret graphs are worksheets where they are actually finding information on the graph but not looking at the whole picture.  They don't address questions of why would someone want to create this type of graph?  Actually seeing the whole picture.   Back to the pie chart and percentages of spending.  Aside from giving lending companies a good idea of if you meet certain criteria, one can also determine if there are areas you can cut back on spending so more money is saved.

I think its also important to provide realistic sets of data from NOAA, scientists, the Census Bureau and other places so students can actually create their own graphs and interpret the results because that puts practice into more real life scenarios.

I plan to do more of this next year in classes.  Its important to have students draw conclusions based on the information presented.  I'll return to this topic later with places you can obtain real data from to use in the classroom.

Let me know what you think.  Have a good weekend.

Thursday, May 11, 2017

Thinking Aloud in Math.

Sculpture, Bronze, Meeting In The Circle  I've read it's important to demonstrate thinking aloud in math so students learn the thinking process behind solving math problems.  This is something they teach instructors to do in certain reading programs but I've never learned to do it in math. 

In math you pretty much assumed you would follow the steps your professor put on the board and not worry about your thinking.  Now, its important because too many students reach high school without thinking how to approach a problem.

When students think aloud, it helps them talk through the details of the problem, the decisions made towards solving it, and their reasoning.  One aspect of thinking aloud is that it makes students slow down and think about their problem solving choices so they have a chance of more fully comprehending the problem.

For students who struggle, thinking aloud can help clarify ideas, identify what they know or don't know, and learn from others when thinking is shared.  This also helps teachers monitor student thinking.

There are a couple of key points to keep in mind when thinking aloud in math.  First is to have students focus on one step at a time while taking their time to fully understand the problem before trying to arrive at an answer.  Second is have students talk about what they notice, the decisions they make, and why they believe their choices are correct.

Some of the ways to encourage thinking aloud include:

1. Model it by explaining your choices, decisions, etc as you demonstrate solving the problem.  Use technology to support note taking, and create visualizations of the bigger picture to help students develop understanding.

2. Provide students with a series of prompts such as I know......, One thing I can try........., or I want to try ........   because......... as a way of helping guide them through the think aloud process.

3. As students become more familiar with thinking aloud, have a peer listen to their thinking and comment on the content while the teacher listens to the strategies being used.

In addition, it is suggested teachers create webcasts which show how different students solved certain problems so students can see how peers solved the same problem.  Furthermore, if a students think aloud is recorded, the teacher and student can listen to it and discuss the process to help the student improve their understanding of the processes used to solve problems.

This is just a scratching of the surface.  I plan to revisit the topic in a few days but its a good start.  Let me know what you think.

Wednesday, May 10, 2017

RTI Part 2

Mathematics, Formula, Physics, School  So back to the question of what is important for high school RTI?

One thing is to make sure students have acquired fluency in their math facts.  I discussed a few reasons for this in yesterday's blog.

According to a presentation on RTI, developing math fact fluency is important for the student understanding of higher level problems.  In addition, they can internalize the math and manipulate numbers in their head, increasing their intuitive understanding of properties such as the associative properties.

The three essential elements of student learning include:
1. They are offered academic opportunities to respond by answering a question posed by the teacher or solve a word problem.
2. They actively respond such as completing a problem assigned by the teacher.
3. They receive timely feedback on their work.

These four goals should be established in any high school RTI program.
Goal 1: Create a supportive math instructional environment.  The classroom should include explicit instruction, use accommodations, and provide strong core instruction.

Goal 2: Develop classroom formative assessments so as to adjust instruction to meet the needs of students.  Why teach one student their core math facts if they already have them down?  If the teacher uses the results to individualize instruction.  One way is to create modules so students only need to study the areas they are weak in, rather than reviewing material they already know.

Goal 3: Develop a menu of math interventions which can be used to better meet the needs of students.  Not every intervention works well for every student. 

Goal 4: Motivate the student to take charge of their learning because the self-motivated student is more likely to succeed.

By meeting these four goals, students are more likely to advance further, faster.  You will see a decrease in misbehavior because they are no longer bored studying material they already know, effort and attendance is likely to increase, success in higher math classes increases, and they are more likely to graduate.

Another aspect of RTI is to give students a chance to learn to think aloud while they work.  One reason for this is students who can verbalize their thinking tend to do better in math because they are clarifying, reflecting, and focusing on completing each step, one at a time.  It also helps them identify what they do and do not know and it slows them down so they check their overall understanding.

Later in the week or next week, I'll spend time on how one teaches "Thinking Aloud" in math.  Let me know what you think.  Have a good day.

Tuesday, May 9, 2017

Use of Calculators and Math Fact Fluency.

Chalkboard, Math Problem, BlackboardOne important facet of math is that students need to have  fluency of math facts.  I realize with all the calculators around, there is debate on if students still need to know their math facts.

Math fact fluency is defined as the ability to quickly recall basic facts for all four operations, addition, subtraction, multiplication and division. As they become more fluent, there are changes in the brain and this ability goes into long term memory so they can focus on other parts of solving equations.

In addition, this fluency allows students to focus on learning problem solving, new concepts and skills which are needed for higher level mathematics.

These math facts are the building blocks for higher level concepts so if a student is fluent, they will find these concepts easier to learn.  In addition, it helps build their confidence in their ability to do math.

According to one article I read, the use of calculators can hinder students who do not have math fact fluency will it can help those who are fluent.  It has been found that having students use a calculator to check their work is much more effective because it provides immediate feedback.

One of the most compelling arguments I've read for having students learn their basic number facts is if they make a mistake entering the numbers in, they know the result is off.  The same applies to entering more complex problems in the calculators, if its not put in correctly, the answer will be wrong.

In addition, knowing math facts is good when working with fractions because you have to multiply or divide fractions with unlike denominators to get a common denominator.  Very few calculators are easy to work with fractions. Sometimes, they give a decimal answer and other times, it is a pain just to enter the fraction.

Not knowing their multiplication and division facts makes it more difficult to identify fractions as equivalent because they don't know multiples.  In addition, this carries over into decimals and percents so overall these students struggle in algebra and higher level math classes.

Furthermore, having your multiplication facts down cold makes it so much easier to list possible factors when factoring trinomials into binomials.  If  you don't know them, you won't be able to factor properly. 

Most people who use calculators use them for more complex mathematics.  I recently saw Hidden Figures which illustrates this point.  The IBM produced two different results for John Glenn's landing spot and they called in the woman to double check the figures.  She did the simple calculations on a fancy adding machine but she knew her math facts so incorrect answers did not slow her down.  She knew.

Let me know your opinion on knowing math facts fluently.  I'd love to hear.  I'm still struggling with the use of calculators because I know my math so well, its often faster to do it by hand than with a calculator.

Monday, May 8, 2017


School, Book, Science, Physics, Maths  I have a superintendent who is very evidence based results. The only problem is that it took till earlier this year for the district to get a school wide assessment which included high school.  We've had others but not for the high school.

We know we have students who are missing key components in their math foundations so they struggle with high school level mathematics.  I will be teaching a foundations of math class next year for these students.

Since its basically a Response to Intervention (RTI) course, I needed to more about applying it to these students.  I've attended a few RTI things but they've been geared more for reading than for math.

I realize this should have started in elementary school but the local elementary school has focused on other things so many students slipped through the cracks mathematically. High school is a bit late but if I can help them become better, that is great.

After a bit of research, I found a lovely report which provided some great information for this topic.  According to the report universal screening, explicit and systematic instruction methods, and data based decision making are the 3 most important practices applied to mathematical intervention.

There is evidence for having students practice fluent retrieval of basic mathematical facts at least 10 minutes a day.  In addition, it is recommended research based programs be included using technology to help engage students.

Another paper, lists 6 reasons students are unmotivated in class which is one reason they may need intervention.

1.  They are unable to do the work due to lacking certain basic skills.
      Students need to be taught the missing skills in order to do the work. One way is to make sure the material in the lesson is is matched to the students ability.  One should open the lesson with a brief review of basic concept or material previously practiced. State the goals for the current lesson, and break all material into small manageable steps.

The other 5 reason can be countered through the use of various methods such as improving lessons to make them more engaging,  assign a choice of problems so students have more control, etc.

2.  The response effort is seen as too big an effort. 
3.  Classroom instruction does not engage student attention.
4.  Student does not see adequate payout for doing the work.
5. Student lacks the confidence they can do the work.
6. Student does not have a positive relationship with the teacher.

In a couple of days, I'll look more at specific RTI techniques for high school math as that is one we often have to deal with as teachers.  I have students who are absent 30 to 60 days a semester so they are missing out on vital information.

Let me know what you think.  I'd love to hear.

Friday, May 5, 2017

Computer Based Testing

Notebook, Laptop, ComputerThe world is changing and has been changing quite a lot in terms of testing.  Most tests are now computer based so the old fall back advice such as "Do all the ones you know how to do first, then go back and do the ones you sort of know and save the ones you have no idea how to do for last."

Not all computer based tests allow that.  So what advice is there for people taking computer based tests?  I checked that topic out so I can share it with you.

Lets look at multiple choice questions because those are the most likely type you will end up taking.  Usually there is one correct and three wrong answers of which two wrong answers are based on the most likely wrong answers testers will have.

There are two types of computer based tests.  The first is the computer based test (CBE) which is just a paper test that has been computerized so you can move through forward or backwards as you need to.  You select an answer rather than bubble so there is a lot more freedom.  The other is the computer adaptive test (CAT) where the answer from one question often impacts the next questions.  In other words, your answer determines whether you get a harder or easier question next. 

In a CBE, you can go skip and go back to unanswered questions while you cannot do that in a CAT where you must always move forward.  Here are some things students can do to help them have better results.

1.  Use scratch paper.  Scratch paper allows you to jot down information and have it all in one place.  Sometimes the tester does not have all the information in one place on the screen so you don't have to scroll up or down.

2. Study the sample test before taking the real test.  Most every computer based test provides small sample tests that can be taken before the big day. These allow testers to have a chance to become familiar with the test so on the day, the tester has less anxiety.

3. If the test is timed, it is important to pay attention to the clock so you don't spend too much time on any one question.  If its not timed, this is not a problem.

4. When you take a practice test, take it with a ticking clock so you get used to the clock being on the screen.  By doing this, you eliminate a potential distraction on the day the of the test.

5. Work hard on getting the first 10 questions correct because some tests put the most emphasis on those questions.

6.  Research how the test scores the questions so you know that ahead of time.

7. Pay close attention to the directions.

8. Options with absolutes such as "Always" or "Never" are usually incorrect.

9. Get enough sleep the night before.

These are the current recommendations for taking computer based tests.  As you can see, the type of test determines whether you can go back or must forge a head.  It seems that more and more tests are becoming adaptive rather than straight computer based so one must pay close attention to the questions.

Let me know what you think.  I'd love to hear from others.

Show Thinking.

Puzzle, Share, Skyscrapers, City  I have done a lot of reading recently where it is suggested students explain their thinking as they solve problems.  This idea is based on two ideas.  First it is easy to find an app or calculator to provide all the steps and solution for any problem out there.  Second, if a student can explain their thinking, it means they understand what they are doing.

Unfortunately if I asked my students to explain their thinking without gently guiding them into the process of knowing how to do it.

So how does one gently transfer explanations from the teacher to the student?  I've actually thought about this and wondered how I could do it in a safe nonthreatening way.

The ideas I've come up with but have not yet tried because we are in our last 10 days of school and things are crazy but I plan to set these up over the summer for next year.

1. Create a matching activity with the steps in order and students have to match the reason to the step.

2.  Create a matching activity with the explanations in order but with the steps mixed up.  The student has to match the step to the reason.

3.  Create a matching game with all the steps and explanations mixed up so students match the step with the explanation.

After they've played with these three matching games, I believe they will be ready to write down their thinking to the problems provided.  Since I work with ELL students, they will need this scaffolding in order to learn what I mean when I ask them to explain their thinking.

In addition, the explanations could be expressed differently each time so students see it done with more than one way of explaining.  This is important because many students get used to seeing problems done in only one way such as y = 22x + 5 rather than 22x + 5 =y or y-5 = 22x.

Many times students will explain each step using a different thought process than the teacher or the book and its important to encourage them to use their own words rather than trying to reproduce the book's explanations.

Let me know what you think.  Do you have any ideas for this?  Drop a comment if you do.

Thursday, May 4, 2017

Should Students Assign Their Own Homework?

Study, Books, Learn, Read, Education  I am aware there is a raging debate out there on whether homework is effective.  I've been in classes where they've said yes it is if you only assign 10 problems and supply the answers. Other teachers are saying it should never be assigned but I stumbled across this article asking "Should students assign their own homework?"

In the past I have assigned only 10 problems based on previously taught material.  I posted the worked out problems so students could copy is they desired.  I've never been one to assign 27 of the same problem.

This article presents the idea of helping students become active in their pursuit of learning.  Taking a step further in understanding that learning is not restricted only to the classroom.  The idea is that students learn they need to make a personal commitment to learn both in school and at home. 

Homework for a student should be designed to help clarify areas they do not understand or is an area they are interested in. If they assign themselves the homework, they practicing self-motivation, preparation, and persistence while evaluating personal growth.

We know you cannot just tell students to assign themselves homework.  They have to be taught how to determine what problems to assign.  In addition, students must see learning as active. So here are some ways to accomplish it.

1. Sprinkle questions throughout the lesson designed to encourage reflection and determine points of learning.

2. Ask questions which can slow down the learning process long enough for students to reflect.

3. Talk to parents to get them on board with the idea.

I think I might take this a step further and use journaling as a way to help identify areas where students still have confusion or have an area they want to explore further.  Once this is done, students then look for problems that cover these areas for the assignment. 

However, before doing this, I think it would be worth taking time to explain why certain problems are chosen as homework problems.  Take time to discuss the best way to choose problems that help clarify.  Perhaps even allow students to use an app or program which shows the steps and have them explain what is going on for deeper understanding.

Is it possible to have students assign their own homework?  It might be but it takes teaching them how to do this so they can make good choices for homework.

Let me know what you think of this?  I'm curious because my first instinct is a resounding no but its so radical, I really don't know.

Wednesday, May 3, 2017

How To Bake Pi.

Antiquariat, Old Cookbook, Cookbook   The other day I stumbled across the book, How To Bake Pi: An edible exploration of the mathematics of mathematics by Eugenia Chang.  The title alone peaked my interest because I love both mathematics and cooking.

Contrary to what you might think, it is both a cookbook and a book on math but the author manages to relate the two.

I admit, I'm not very far into the book but I am impressed with her approach. Each chapter begins with a recipe but the rest of the chapter discusses math in terms of much of the recipe.

Chapter one looks at the question "What is Math".  It begins with a recipe for Gluten-Free Chocolate Brownies.  She equates math and recipes because they both have ingredients and method.  She shows how important it is to have both so the results are good.  I like the way she discussed math being about technique. 

She takes time to discuss how with many of the same ingredients, its the technique which changes the results.  For instance, potatoes can be prepared using different techniques which effect the end result.
Chapter eleven looks at relationships.  It starts with a recipe for Porridge and promptly asks how big is a cup?  She goes on to point out that as long as its all done using the same cup, its doesn't matter the size of the cup because of the relationship.

She takes time to use real life examples of the same concept to help relate it. Back in the first chapter, she discusses characterizing  something by what it does by using the Prime Minister for example.  She goes on to point out that its hard to describe math because once we have a technique we can apply it to more and more things.

Its like beginning with roasting potatoes but you can also roast sweet potatoes, beets, carrots, parsnips, asparagus, kale, and all sorts of other vegetables.  As you find more things to study, you find more techniques that apply.  Kind of like the chicken and the egg.

In chapter two she talks about abstractions which she says are actually blueprints.  Its like cottage pie, shepherd's pie or fisherman's pies are all about the same in they have a mashed potato topping but the filling is different.

I love the way she uses facets of cooking to relate it to math in general.  I usually prefer finishing the book but its currently on sale in Kindle format for $1.99.  Its well worth the money.  Have fun and let me know what you think of the idea.

Tuesday, May 2, 2017

Cut, Copy, Compare

Logarithm, Board, Mathematics, Pay  Yesterday, I discussed three techniques designed to help students learn their multiplication facts.  One of them struck me as a method, I could easily use in my classroom to help students learn certain processes which are applied in a variety of situations.

The method is known as Cut, Copy, Compare where a worksheet is divided into two columns.  The problems are worked out in the left column while the right column is empty.

Students are asked to study the problem with the answer on the left side.  When they think they know it, they cover the problem and write it on the right side, then they check their work.  If they are correct, they may move on to the next problem, if not, they continue to study the problem before trying again.

I've noticed areas where I can do more to ensure they know how to do something but I don't always have ways to help them learn it better but this method might work well with the following topics.

1.  The rules of exponents.  I don't spend enough time making sure they know the rules.  They've gotten these same rules in middle school but I don't believe their teacher takes the time to ensure they really know the rules.  Its important to know the rules of exponents for logs and natural logs since they use them extensively.

2.  Solving one step equations is another area this might work.  Most of the time we have students practice solving problem after problem without taking time to have them study the process.  I've seen time after time where students need immediate feedback so they learn to do it correctly rather than learning it wrong.

3.  Solving two step equations once they've learned to solve one step.

4.  Combining like terms

5. Solving algebraic fractions - review the basic rules for solving fractions with unlike denominators before solving algebraic fractions.

6.  Trigonometric ratios - could include solving to find the missing angle or side, the Law of Sines or Law of Cosines.

These are just a few I thought of, in 5 minutes.  I am due to teach a Fundamentals of Math for students who seem to be missing the basics and struggle.  I see this method as great for having students practice working with fractions or decimals.  My next step is to figure out how to use it digitally so my students do not lose their papers. 

Let me know what you think. I'd love to hear from people.

Monday, May 1, 2017

Multiplication tables.

School, Book, Maths, Logs, Logarithms  Although I allow students to use calculators for multiplication, I worry they have not developed the number sense to know if the answer is in the ball park.

I realize most people use calculators to find the solution.  Even in the movie "Hidden Figures", the women used a mechanical device to do the calculations but they'd learned their multiplication tables so they knew if their answers were reasonable.

Knowing their multiplication facts falls under the general topic of Math Fact Fluency.  It has been found students who are fluent, have also developed strategies and use them to solve these problems.   In addition, these students have more cognitive resources available to learn more complex concepts and complete complex tasks.

Furthermore, being fluent means students experience less anxiety and increases their confidence so they are willing to solve more complex tasks.  Math fact fluency means a student can spend more time solving the problem rather than struggling to work through basic computation.

It has been found, the best interventions have components of practice using modeling and drilling  so students see how to do it and practice doing it.  Modeling could either be student or teacher based but there are three interventions which come highly recommended.

1. Taped-Problems Intervention in which the problems are recorded in audio form with consistent pauses between the problem and the answer.  The idea is for the students to listen and answer on the accompanying worksheet before the recording gives the answer.  If they have not written an answer down by the time the audio answer is given, they write it down.  If their answer is incorrect, they cross it out and record the correct answer. 

It is suggestion, that as students become more proficient, the pause becomes longer so they have time to record the answer.  This method provides immediate feedback.  Be sure to create at least three different recordings.

2. Cut, Cover, and Compare has students look at worksheets with the problems and answers on the left hand side but with the right side blank.  They are to study the problem and answer on the left.  Once they think they know it, they cover it and write it on the right side before checking.  If the answer is correct, they move to the next problem.  If its incorrect, they study the problem again until they get it right.

3. Incremental Rehearsal - Assess which facts the student already knows by giving a short test or activity.  Create flash cards using 9 facts the student already knows and 10 facts the student does not know.  Show the student the first unknown and ask him or her to answer it.  If they answer incorrectly, give the answer, then give them a fact they know to answer.  Anytime he answers an unknown incorrectly, give them two known facts before having them attempt another unknown. 

Once a student can answer the unknown, it becomes known.  Repeat until they know all the facts and test again to find the next facts they need to learn.

These are three suggested ways for students to learn their multiplication facts but I see a great use for Cut, Cover, and Compare in High School Math.  I'll talk about it tomorrow.  Let me know what you think.