Improving Algebraic Understanding

 I have been at a loss figuring out how to help my students learn and retain their ability to understand and work out Algebraic problems.  One day, they can do it, the next they act like they are reading Greek.

The other night I came across an article focusing on improving algebraic understanding in middle school and high school.

The paper has three major suggestions for ways to improve algebraic understanding and one of the ways mentioned, I've never actually heard but it makes sense.  After reading it, it made so much sense but I also know I have to teach my students how to do it.

The first area to concentrate on is analyzing solved problems that show all the steps.  The idea is they look at the steps and discuss them to find the reasoning and strategies used.  In other words, they are finding connections.  When they find these connections, it makes it easier for them to transfer the information.

Furthermore, the selected problems should apply to the objective of the lesson and some of the examples should include common mistakes so students learn to watch out for them and to look for them in their own work.  This should be done using whole groups, small groups, and independent practice to teach them how to do this.

One big reason for showing solved problems is it allows students to see the the strategy used in total context rather than each step.  When they discuss the reasoning behind the steps, it helps students develop a deeper understanding of the logic involved in solving the problem.  Looking at incorrectly done problems help students learn to think critically.

The second area is to teach students about mathematical structure and algebraic representations by using proper language, teaching them a self reflective method they can use to notice structure as they solve the problems and help them understand that different algebraic representations provide information about the problem.

This process helps students connections between problems, strategies, and representations that at first may seem different yet turn out to be the same.  When they understand the structure of problems, they have time to focus more on the similarities between problems.  In addition, if they understand what makes an algebraic expression an algebraic expression, they will see the connections regardless of how they are presented.

The last area is to work with students to choose from several strategies to solve problems.  It is suggested students make a list of strategies they can use to solve a problem, explain why they chose the strategy, and they need to evaluate a variety of strategies they could use to solve the problem.

Its important to teach students that strategies are different from algorithms.  Algorithms are a set of steps one always follows to get a specific result where as a strategy requires students to make a choice based on the problem and the end goal.  Getting them to move past the algorithm means they can move past memorization to true understanding.

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

Universal Design Learning in Math

  We are always looking for ways to improve our teaching so students learn the material better. One way to do that is to utilize Universal Design for Learning or UDL when planning lessons.  UDL allows for more flexible ways to teach lessons and more ways for students to show they've learned the material.

Behind the UDL is that when used, it can reach all students.  There are three basic principals UDL is built on.  First one must provide multiple ways of representation, second provide multiple ways of actions, and third provide multiple ways of engagement.

The teaching practices that meet UDL design includes video captioning, multiple online resources, observation, organize your board, and use multiple forms of assessment.  Some of the recommended ways to organize your board are to cover the board left to right, underline important information, recap the previous day's material, and always include the outline of a schedule for that day's lesson.  My schedule usually looks something like warm-up, notes, practice problems, and a game.  Sometimes instead of notes, I might use a video or even use a video for practice problems.

One reason for making sure the video captioning is on is to help students focus on the material instead of just watching it.  I know for myself, I always put the closed captioning on so I can read the what is being said in addition to listening to it.  I find, I pay more attention to what is going on.  It also allows hearing impaired students a chance to follow the videos without feeling as if they are different.

As far a online resources go, there are so many sites out there filled with video's, tools, examples, quizzes, visual aides to help students.  Some of these can be assigned to help students do the work at home.  There are multiple types of online calculators students can use to check their work for various types of problems.

When. observing students, watch for facial expressions indicating their confusion, ask questions, make them feel as if you care about how well they do.  Not every student is going to ask for help because they might be too shy, too insecure, or too lost.

So you may be wondering what a UDL lesson plan should look like.  The teacher should anticipate student needs when building the lesson plan.  When creating the lesson, build in differentiation so the need of each student is met.  The lesson itself should be filled with well supported activities which engage students in a safe learning environment.  Include multiple ways to learn and show knowledge and skills.  Assess student learning and adjust often.

Some ways to increase engagement include using alternate seating, offering choice boards, giving brain breaks, think-pair-share, peer tutoring, timers, and exit tickets.  Some ways to represent the mathematical concept would be a Three Act Math activity, visual cuing, text to speech options, manipulatives, word walls, or flash cards.  Suggested ways to show what they know might be using online tools, acting something out, record a video, use manipulatives, make an oral presentation, or think-pair-share.

This is just an introduction to give people an idea of UDL and its application in math but I'm planning on addressing this topic later on in more detail.  Thank you for reading this.  Let me know what you think, I'd love to hear.  Have a great day.

The Mathematics of Game Shows

My mother loves watching Wheel of Fortune while my father is a Jeopardy lover.  Both play the game from the comfort of their chairs hoping they beat the contestant with the correct answer.  I figure its good for their brains because it exercises the brains and keeps them active.  When I watch, I think of the odds each player has of winning the game.

I'd love to teach a statistics class focused on the odds of winning various television games, especially the more popular ones. Most every game on television is based on the statistics of spinning the right space or choosing the right letter or number, or space.  All game shows are based on probability.

If you've ever seen Let's Make a Deal, you know they have one activity in which a person has a choice of selecting one door for a prize.  If they get the goat, it's over.  So the person chooses a door and the MC shows the goat  behind one of the other doors, indicating that one of the two left has the prize.  The person is then given the choice of keeping the door they chose or switch to the other one.  Most people stay with their choice because they feel they have a 50% chance of being right but mathematics indicates, it is better to switch because it increases your odds of winning.

I found a paper that sprinkles game show probabilities with information on regular probability. Its actually notes from a class but I like the way the author discusses various games throughout the paper.  The first two shows he discusses are Deal or No Deal, and the Price is Right with a short summary of each game.

As he introduces probability beginning with the basics such as sample space, followed by  expectations, counting, poker, inference, competition, backwards induction, and special topics, he sprinkles bits about various game shows and how certain parts apply to the topic.

For instance, when he introduces sample spaces, he discusses the squeeze play in the Price is Right.  In this game, you remove one digit, usually the first or last, to find the correct price.  He shows how the sample space applies to this particular situation.

As the author discusses various topics, he continues to apply probability to each situation which makes it more interesting because its not usual examples.  Back to the Price is Right and its 10 chances game.  The small prize has a two digit price but the contestant is given three digits,  the medium prize has a three digit price but given four digits and the big prize is made up of five digit price with no extra digits.  The object is to guess the correct price to win as many prizes as possible.

The author analyzes the probability of getting all the prizes and how to figure out the most likely prices based on mathematics.  The cool thing is that every example he sites, he's included a video clip so you can actually watch the activity he's talking about.

This 146 page paper is actually class notes form a class taught on The Mathematics of Game Shows from 2018.  I had a blast reading it but I plan to go back and read it in more detail.  Let me know what you think, I'd love to hear.  Have a great day.

Game Templets

 I try to integrate games into my classroom as I can because my students would rather play a game than do anything else.  I think part of it may be due to the fact they all play a variety of games on their devices outside of school.

I've been using Kahoot and Jeopardy a lot but there are other games I can use.  The other day, I stumbled across a list of templets for a variety of games I could easily use in the classroom.

I"m sharing sites you can make power point type versions of several different game shows.

One of my favorite sites for finding Jeopardy games is at JeopardyLabs.com where I could also make a game if I chose but they have tons of done ones.  On the other hand, you can use power point templets to create Jeopardy games on your computer.  This site lists 12 places you can go to find templets to create your own games including one that uses Google Slides.

To shake things up, how about a family feud game for a change.  It uses two teams with 5 people on each side.  The question is asked and people answer.  It is possible to make one for math but you need the templet to create a game.  If you check here, you'll find 6 sites which will allow you to create one for your math class.  Each is a bit different but they all work about the same.  The one game I saw referred to as Family Feud, had students answer questions and show their work in order to get the point.

If neither Jeopardy or Family Feud is something you like, you could try Wheel of Fortune.  I realize it works mostly using words but it could be used with mathematical vocabulary.  This one is considered quite realistic while this one is definitely a power point presentation set up as a Wheel of Fortune Game.  I think it would be a great way to practice vocabulary, operation words, even mathematicians.

I've often wondered how one could use The Price is Right in the classroom but it's taken me a while to figure that out.  Its a great avenue for having students practice cost per ounce for various items.  Rather than giving a product to students to guess the price, why not provide two versions of the same product and have students calculate which one is the better buy?  This site offers a power point with 8 slides that could easily be edited. This one on the other hand, is simple and has only one set of slides but you can easily make copies.

If you've ever seen the Cash Cab show, there is even a templet for creating one of your own.  The templet is set up for 12 questions which could easily be math questions of any level.  Students need to work through the problems before they come up with an answer.

Last but not least is the "Who Wants to be a Millionaire" with three different templets.  The first has space for 12 questions, easily edited.  This Google slides version is ready to be edited for your topic while this one comes with music, lifeline, etc.  The questions can be set up from easy to harder to hardest which is the way normal games are set up.

So all these templets are downloadable and you can create multiple games to use in your class.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

Mars is 140 million miles from the Earth.  If it takes 3888 hours to fly to Mars, how fast is the space ship flying per hour?

Warm-up

The moon is about 238,900 miles from the earth. It takes four days to get to the moon. What is the rate of travel per hour?

Matching in Math?

Note: Sorry, today's column is running a bit late but due to attending a conference, traveling back in time for Parent Teacher Conference, I didn't get things done ahead of time.

Back to the regularly scheduled column.   Last night, my mind woke me up in the middle of the night thinking about matching games based on the concentration model.  The one where a person chooses two cards to see if they match.  If they match, the person gets the pair, if not they are flipped over for the other person to try.

The conference I just finished attending, said that sometimes you need to unplug and do things without digital devices.  These can be done using index cards and pens.  You can have students prepare sets so that you do not have to make them.

1.  Vocabulary - Prepare two cards per word.  The first card has the word while the second card has the definition.
           Make a set with the vocabulary word and an example
           Make one with the vocabulary word and a picture,

2.  Word sentences - Prepare two cards per word, the first card has the written sentence while  the second card has the algebraic or arithmetic equation.
           This can be done the reverse way from the equation to the word sentence.

3.  GCF or LCF.  The first card has a pair of numbers while the second card has the GCF or LCM of the two numbers.  You just have to make sure none of them repeat so they do not share answers.

4. Process - The first card can have the equation while the second card has the next step to solve the equation.  Its quite possible to set it up so it takes a couple rounds to solve the equation from start to finish.  If an equation takes four steps to finish, it requires the student to find two matching pairs.

5.  Geometry - the first card has the shape while  the second card has the the name of the shape.
             The first card has the shape while the second card has the formula for area, volume, or perimeter.
             The first card has a description of the shape while the second has a picture.
             The first card has a picture of angles and the second card has the name or vice versa.

6.  Decimals, Fractions, and Percents.  The first card has a decimal and the second a fraction or the first a fraction with the second being a decimal, or a decimal and a percent.

So many possibilities to create games.  If you would rather use digital devices, there are some websites you can go to to create actual games.

This site allows you to create your own matching game but they show both options at once you just have to match them from a crowd of choices.  They also have a gallery of games made by others so you don't always have to begin from scratch.  It is free and allows one to sign in through your google I.D.  It also gives you a code to let people access the game.

This is a site that makes the game more of a concentration with cards facedown.  You can create for free but it costs if you want to download it to your computer to print. 

Both look easy to put together. The only reason, I'm more into the cards rather than the digital for this game is that I can have students talk a bit more rather than focusing on their devices.  Next week, where do you find templets for other games and google home use in the classroom.

Have a great day and let me know what you think.