What Happens When We Don't Understand Fractions.

There have been events throughout recent years that show most people do not have a good unstanding of fractions.  By that I mean that they do not grasp the differences between one-third and one-fourth or five eights and six ninths.  Many ended up with the idea that the larger the number in the denominator, the larger the number which as we know is not really true.

If you look back to when the A & W restaurants tried to take on McDonalds by offering their own burger deal.  They decided to offer a larger one-third pound burger which was bigger, tasted so much better in blind taste tests,  and cost less.  In other words, they offered a better deal than McDonalds but people didn't go for it because they didn't understand that one-third is bigger than one-fourth. People thought that one-third was less than one-fourth because three is less than four.

Another example comes from the 1980's when the game Dungeons and Dragons became popular.  A they spread across the country that players of Dungeons and Dragons were more likely to commit murder or suicide due to a few cases.  This happened during the time when the United States was undergoing a panic about Satan and his influence and it even made the news show "60 minutes". Unfortunately, no one took the time to actually look at the numbers of players who committed murder or suicide versus the total number of people who played the game.

One person researched the numbers and over a five year period, only about 28 players either committed murder or suicide out of the 3 million or more teenagers who played the game.  In order for the fear to be true, the numerator should have been at 360. or more based on the normal number of teen suicides back in the 1980's.  So in reality, the people who played Dungeons and Dragons were less likely to commit suicide than the regular teen age population.

Now if you look at a test given to more than 20,000 eighth graders about 40 years ago, only about 24 percent were able to answer questions on fractions correctly.  Since then, the number of students who showed they understood fractions only increased to 27 percent.  Not much of an increase.  It has been discovered that how well a fifth grader understands fractions and can use them can be used to predict how well they will learn algebra and what they achieve in high school mathematics.  By the time a student gets into high school math, they will be facing more problems and formulas containing fractions. In fact, a lack of knowledge of fractions can impede students learning high school math and can make it difficult to do certain jobs once they are out of high school.

Furthermore, when students learn fractions in elementary school, they often have trouble intrinsically with being about to use what they've learned.  For instance, when they multiply fractions, they have to multiply the numerators and then the denominators but in adding or subtracting fractions, they cannot do that. This and other things makes it difficult for students to learn to use fractions challenging.  

The second thing that makes it more difficult is that teachers often find it difficult to explain why problems are done a certain way.  For instance, many teachers are unable to provide an explanation for why we turn a fraction division problem into a multiplication problem with the reciprocal. It's even difficult to provide a visual to show fractional division.

So when these same students become adults, they carry the same lack of understanding with them so when they see things like the A & W third pounder, they have no idea it is a bigger hamburger.  Humans have a tendency to not look at the full picture when deciding if a situation is really bad or good. Instead, humans rely on a cognitive framework to make quick decisions.  This process is called heuristics and for the most part it works but when it fails, it fails horribly. 

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

Student Created Math Handbooks.

Most students do no want to take their textbooks home and notebooks just contain notes based on what the teacher has provided.  There is something between the two that can help students organize the information into a usable form. 

Instead of giving note after note, have students create their own personal math handbook.  A handbook is a portable reference guide while the textbook is the course book filled with everything and is often quite thick and heavy. The handbook is different than the regular notes in that the handbook is more organized with proper examples and is easier to use because the topics are together.

Often, students are introduced to equivalent fractions before learning how to compare the fractions and a bit later they are introduced to making sure denominators are the same when adding and subtracting fractions.  Many times, they have notes on other things scattered between the topics so by creating a handbook, the student organizes the information into a more cohesive unit.  When they redo the material as they organize the handbook, they are reviewing the information so their brains have another chance of learning the material.

A student created handbook can be made using a composition notebook or an app on the iPad or computer.  The way a student puts the handbook together is the same for either the hard or digital copy.   The first page or two is always going to be the table of contents.  The listing will not be every single topic covered in their notes but will be grouped according to topics such as fractions, decimals, volume, quadrilaterals, etc.  The page number is the page number the section starts.  So it might say Fractions.          7-12 letting the reader know that all the material between page 7 and 12 is information on fractions.

The information written in the handbook may answer the what is it, how is it used, or why is it done this way. It doesn't have to include everything, just the information needed to do it.  For instance, when placing information on fractions, it might include the vocabulary for the numerator and denominator, finding equivalent fractions and why they are needed, how to make fractions equivalent with one or two examples, adding and subtracting, multiplying or dividing fractions, and improper and mixed fractions. This section could cover regular fractions or algebraic fractions.

In addition, each section should have several extra pages so the student can go back and add additional information and examples as needed.  The organization makes it easier for students to use the material and they can identify relationships between topics and concepts.  Yes, this might be adding another level of work but it allows students to look at the most important concepts and ideas.  The student might show solving one step equations with fractions and decimals in their notes but they would show the general formula in their handbook with short notes on what to do if the problem has fractions or decimals.  The student can also include a short note about referring to page 27 in their notes for more in-depth information and examples. 

So handbooks provide a quick organized way for notes so they are in a more cohesive form.  It also allows teachers to help students learn what is the most important information out of their notes and what isn't but is still good to have.  Let me know what you think, I'd love to hear.  Have a great day.

Digital Task Cards

Most teachers end up being out of the classroom at some point.  It's always hard to set up something for students to do when you are out but I've used a learning platform to leave the lesson for the day so I don't have to make tons of copies of make-work for my sub.  Having students use task cards is one option and they can be done digitally. 

Task cards are cards with problems for students to do.  Although most schools are back in session, some students are out due to covid so by having digital cards, all students can participate even though they are not at school.  They can also be a part of the lesson used by a sub when the teacher is out.

Digital task cards can be set up so students can either scroll through the choices or work their way through the cards so the teacher becomes the facilitator. In this case the facilitator can walk among the students, encourage them to work together, edit each others work, explore, analyze, while providing a choice.  Furthermore, the tasks can be differentiated from relatively simple to much harder so students can find something they can do. Digital Task cards allow students to work at their own pace or the teacher can set a deadline.  There can be choices for students who finish early or need something more challenging. 

There are a couple ways to create and use digital task cards.  One way is to create the cards in Google Slides. Open google slides and set up a new presentation.  Place a different task on each slide. One can create four answers on each slide with one correct and three incorrect answers.  Link the correct answer to a slide telling them it is correct and when they click the next button, they go to the next question while the incorrect answers are linked to slides that say incorrect and back.  The other possibility is to create a open ended task that students answer using a google doc where they explain their thinking. Once they have their answer down, they move to the next slide.

If you prefer, you can set up a google form for students to place their answers.  Each slide lists a different task and each task should be numbered.  Then set up a google form so it's set up as a self-grading quiz.  If you go with tasks that have absolute answers, set up google forms with the correct answer so students type in their answers and they find out immediately if their answers are correct.  Otherwise, if the tasks are open ended, set the form to short answers so students can place their answers in the form and you can provide feedback.

Another way to create digital task cards is to take a pdf of all the task cards and capture a picture of each task card.  Then import one task card per google slide so you have a slide presentation filled with the cards so they can choose one or work their way through.  This can also be done in google forms so that one picture of a task card is imported for each question.  In the question box, you type the number of the task card and import the task card image for the question.  Click short answer and required, then done. 

If your questions have a single answer, you can click multiple choice and list four answers for students to select from.  The correct answer is identified and inputed so when students chose the correct answer they get immediate feedback.

If you use keynote or power point, you can set it up as a presentation and import pictures or type out the questions and follow the directions for google slides with the appropriate links.  You can have students type the answers into pages or Microsoft word.  So now you know how to create digital task cards.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

If there are 1100 cinnamon flavored candy hearts in a pound, how many would be found in a 12 ounce bag?

Warm-up

 If there are 137 hearts in a 12 ounce container of jelly hearts, how many are there per ounce?

Ways To Use Task Cards

Task cards are a great activity to use in the classroom as it encourages independence, critical thinking, used for review, assessment, and can replace worksheets.  Task cards can be either in hard copy or digital form. Today I'm just exploring ways to use them in class and on Monday, we'll look at how to create digital versions that can be assigned via a learning platform.

A task card is a card with a question or task students are asked to do or answer.  These are great for reviewing a topic or concept and they are not a good activity for introducing something new. 

Task cards can be used in so many different ways in the classroom.  Rather than seeing them as "tasks", see them as a versatile element of instructional activities. It is important to know what process students are using the cards for.  It is a review, additional practice, or assessment so you know what the expected result is.

Task cards can be used with individual students to reinforce their learning of a specific concept or topic if they need a bit more time to master it.  In addition, a teacher can use the student's answer to assess their understanding of the material. Students can do a many as needed for them to learn the topic so if they just need one more time, they do one task card, if they need several more tries, they will do those.

Instead of sending home a worksheet or book assignment, use task cards so they an work on the skill outside of the classroom.  This is a perfect situation for either a physical or digital task card.  The advantage to using a digital task card is that the student cannot forget or lose it.

For students who are ahead of everyone else, create more challenging task cards so these students are not bored. These can be available to those students who always finish early or need enrichment.  Task cards can be created for students of different ability levels.

In addition, physical task cards can be used to create movement within the classroom.  It often helps students focus better if they can move around the classroom. Just make sure students know what you expect when they move around.  One way to do this is to place task cards around the room like you do in search and rescue and have students work their way through the cards.  They write their answers on an answer sheet where they can show their work.  At the end of the period or time, they turn in answers to the teacher.

Task cards can also be used in small groups by passing out one card at a time to each group and have all the members of the group write their answers on white boards so they can compare answers.  If any of the answers are different, it opens up the group for a dialog so they can work on determining which answers are correct or reasonable.

Turn the use of task cards into a game such as "Scoot".  In this game, students each get a stack of task cards. They need to answer as many as they can within two to three minutes. They can write answers in their notebook, on a whiteboard, or in a journal.  When time is up, the teacher calls "Scoot" and the students either move to the next stack to the right, or they pass the stack of cards to the right and they do it again.   Another possible game is to have two students, standing back to back,  in the middle of the room.  They are each given the same task card that the teacher reads the card so the class knows what they are doing.  Students write the answers on a whiteboard and when they are both done, they compare answers and then two new students are chosen.  

It is possible to turn the material on task cards into a board game that students play.  Task cards are ideal for interventions.  Use them when working with students who need additional help.  So many ways to use task cards instead of worksheets.  Let me know what you think, I'd love to hear.  Check back on Monday to learn how to make digital task cards.   

Adaptive Learning Software.

I'm sure by now, most teachers have used adaptive technology in their classrooms.  This past semester, I had students working on a program that used adaptive technology but they didn't understand that when they used calculators to do the arithmetic, it would give them harder problems because they gave correct answers.

Adaptive learning software is software that is used to teach and help the student learn via personalized lessons, practices, and assessments. It does this by using artificial learning and machine learning techniques to "adapt" a learning plan for individual students.  It also provides data for instructors and the administration to analyze so they can provide instruction to meet the needs of the students.  Usually, adaptive learning software is web-based and contains all the information necessary for the class so it can direct students along their journey of learning.

Adaptive learning software works by assessing what skills or concepts students have mastered before adjusting lessons or activities so students learn.  So this software begins with an evaluation to determine which skills or concepts the student needs to work on, creates an individualized pathway, and monitors and assesses their progress.  Some programs can distinguish between performance data and engagement data.  In other words, are they really engaged in their learning or just there going through it.

Adaptive learning software is able to provide a plan for personalized learning for every student with scaffolding, flexibility, targeted learning lessons, and any resources needed. In addition, they provide immediate feedback on practices, clear learning objectives, determine what is needed to improve, and low stakes assessment.

Furthermore, the software allows students to monitor their progress so they are more self-directed, work at their own pace by skipping skills and concepts they've already mastered while reviewing and practicing new concepts and skills more slowly than with direct instruction. It provides lessons that match their needs and readiness, can be used to replace textbooks, provides a framework of lessons, activities, instruction, and assessments that meet their needs while allowing instructors to provide additional instruction to students for areas they struggle.

Since the software contains all materials needed for the class, it is able to break down the concepts and skills into properly sequenced and manageable chunks leading students to mastering the learning goal. The software works on providing instruction and activities between the students comfort area and the zone where they get frustrated which is known as the zone of Proximal Development. This zone is the area between what students have already mastered and the material that is too challenging so they get frustrated.

Many schools are relying on adaptive learning software to help fill in gaps of student knowledge gained during the pandemic. Based on my personal experience, the students who really work through the material and try, tend to learn but those who didn't try, didn't accomplish much.  Let me know what you think, I'd love to hear.  Have a great day.