Mathematical Thinking.

 

The third part of teaching for mastery is to help students develop mathematical thinking.  This is the process where students look for connections, reason, generalize, and so much more.  Mathematical thinking is also referred to as mathematical reasoning. Although it is better for students to develop this skill in elementary schools, we can help middle and high school students learn it. 

So today, we'll look at ways to help students either improve or work on their mathematical thinking/ reasoning.  It is important for students to have strong reasoning alongside the skills to learn and do well in mathematics.

In fact, mathematical reasoning is often thought of as applying both logic and critical thinking to a math problem to make connections so as to solve it.  It is what bonds the students skills together and connects fluency to problem solving.

There are two types of reasoning, one is inductive and one is deductive.  Inductive reasoning is where one comes to a conclusion based on what they observe so the conclusion may or may not be factual, Deductive reasoning is based on reaching a conclusion based on facts.  Over all, mathematical reasoning requires the use of mathematical vocabulary and active listening.

It is suggested that one start class with a question or activity that requires students to struggle.  In addition, you as the teacher are not the answer key so don't provide answers.  Let students share their original ideas. The question you pose at the beginning of the class could be the objective just rewritten into a question.  Instead of telling students that they will be learning about using a ratio to describe the relationship between two quantities, ask them what it means if you use a ratio of fat to flour of 1 to 2.  

When you change learning objectives into questions, you automatically engage students because they have something to think about.  In addition, they have to show their thinking or reasoning when they find an answer.  When they have to share their thinking, they begin to develop deeper understanding of the material.

Rather than providing hints or answers, help students learn to unpack the question at the beginning of the lesson so they bring their ideas together during the lesson.  This leads to a mathematical discussion where they may have to provide visualizations to show their thinking.  Think about using a "never, sometimes, always" activity to help foster reasoning and problem-solving skills.

So think about starting the lesson with a question, a provocative mathematical statement or mind bender.  Present answers as puzzles rather than just giving the answers. Group students together in threes so one talks, one records, and one listens and watches.  Sprinkle reasoning prompt posters around the room.

Have fun trying some of these ideas.  Let me know what you think, I'd love to hear.  Have a great day.

Representation And Structure In Mastery

 

Representation and structure is the second over lying idea when discussing teaching mastery in math.  It is easy to learn the big ideas but it can be much harder if you have no idea how to use it in your classroom.  Today, we'll look at ways to incorporate  representation and structure into your classroom.

Representation and structure refers to the different ways in which mathematical topics or concepts can be represented.  In addition, these representations can be used to show how certain concepts or topics are connected.

For instance, one can represent an equation concretely, pictorially, or abstractly.  Concrete refers to using manipulatives to represent the equation or problem.  Pictorially refers to drawing pictures to show the idea while abstractly uses the symbols or written form to take it a step further. Abstract is also known as  Students need this because we have discovered that there is not a single way to do any mathematical equation.

The thing is that the concept or topic needs to be presented in the correct order.  The best way is to begin with a pictorial representation so students "see" what it looks like. I will admit that this can be more difficult in higher level math classes but it can be done.  Once they've explored and gotten the pictorial representation, then its time to move on connecting this to the representation in the text book.  Finally, connect these to more abstract view of the concept by connecting the idea with prior knowledge. 

It is important for students to use language, written words, and reading mathematics is extremely important to learning mathematics.  This is part of the abstract stage and students have to be able to read and understand the textbook.  

As far as representations, there are so many possibilities to use, even in middle school and high school maths.  There are base 10 manipulatives, math balance scales, and so many more.  In fact, manipulatives can be found either in virtual or manipulative form and some are free, some do cost but if you look around, you can find things.  One can even find free apps for student devices so students can use their phones or computers.  These manipulatives can really help students make connections.

Next time, we'll look at mathematical thinking in more detail.  Let me know what you think, I'd love to hear.  Have a great day.

Warmup

If there are 350 worms in a pound, how much does each worm weigh?

 

Warm-up

 If a pound of gummy worms contains 58 gummy worms, how much does each worm weight.

Coherence In Teaching Mastery.

Last time I briefly touched on the 5 big ideas when it came to teaching for mastery.  Now we'll look at each topic in more detail to see what each is and how to help students become better at it.  Today, we'll look at coherence in mathematics and ways to help students learn and use it.   

Coherence is the idea that all big mathematical ideas are connected to each other.  When topics are introduced so students see connections between what is being taught now and what they've learned before, they are more likely to do well.

Mathematics should be taught in a logical progression so if they learn to multiply a whole number by a fraction in fourth grade, they have a proper foundation for multiplying a fraction by a fraction in fifth.  Coherence allows students to understand the topics to a much deeper degree while they are better able to grasp the more complex topics.

If students have not learned a previous skill, there are opportunities available to sneak it in so students get the chance to learn it.  For instance, use the missing skill in a warm-up that connects to  prior learning.  Another possibility is to build prerequisite instruction into a grade level lesson, or actually teaching a lesson on the previous skill before teaching the actual grade level lesson.  

Take time to monitor the progressions across grade levels so you know what skills they should have learned in the past, in what order.  With this information, one can look at the skills you might have to include in your instruction so students will be successful. 

Then begin the class with an activity that builds on a previous skill that relates to the topic for the current lesson. This helps students be successful and it highlights how what they know already, relates to what they are learning today.  When teaching a lesson, allow students the opportunity to explore the current topic to see how it relates to what they've learned previously, by using rich tasks that have multiple entry points. In addition, check for student understanding of the current lesson before providing the opportunity to practice and experience differentiation via reteaching, reinforcement, or enrichment.  

Next blog entry on Monday, I'll look at representation and structure.  Let me know what you think, I'd love to hear.  Have a great day.

The 5 Big Ideas In Teaching For Mastery

 

When we teach, we are teaching students with the idea that they will master the material.  Sometimes, we need a reminder of what it involves and ways we can manage it while other times, we stroll along.  For many of us, school has started and we are doing the first of our assessments. So today, I'm looking at the overall picture of teaching for mastery.

First is coherence which is the way topics are taught so they create a coherent learning progression throughout the grades.  They hopefully will end up with a deep understanding of the concepts and topics so they apply math across a large range of contexts.

Second, is carefully selecting representation of the concept or idea to show the structure.  This results in students "seeing" math rather than using it as a way of doing the math. The representations become mental images for students that they can think about as they work on the math and which helps them develop a deeper understanding.

Third, encourage and help students develop mathematical thinking which allows them to make connections, develop conjectures, perform reasoning, and generalize.  When planning any lesson, students should actively be involved in thinking mathematically and communicating using mathematical language. 

Next, all students should have developed fluency of number facts and use of operations so they can spend more time learning the concepts. In addition, fluency also means that students have the flexibility to move between various contexts and representations, be able to make connections as they recognize relationships and choose the best method to solve a problem.

Finally, students need to develop both conceptual and procedural variation.  Conceptual variation is when students are able to vary the way a concept is represented to show a critical feature.  Procedural variation looks at how students "proceed" through a problem. 

These are the 5 big ideas in teaching for mastery and are an overall look.  Over the next few weeks, we'll look at each idea in more detail and see how they can be applied in a post Covid world.  Let me know what you think, I'd love to hear.  Have a great day.

Mathematical Modeling On Body Clock Distruptions

 

It happens to everyone.  Their sleep patterns are disrupted due to time changes through daylight savings time, jet lag, working night shift, or playing on your phone into the early morning.

Researchers from two universities created a mathematical model to help explain the resilience of the body's master clock found in the brain.  There is a cluster of neurons in the brain that work together to manage the internal rhythms of the body.  They also hope the results lead to suggestions on how to improve the internal clock in people who do not have a good internal rhythm.

The reason for this study is because continued disruptions to a person's circadian rhythm can cause diabetes, memory loss, and other issues.  Since society has created a situation where many people end up working hours outside of the normal day light hours, it can be be hard on being exposed to light, eating, and sleeping.  

The researchers created a mathematical model that looked at the suprachiasmatic nucleus (the internal clock) as a macroscopic or big-picture system made up of an infinite number of neutrons.  They looked at the connection between the neutrons in the system and how they manage to have a shared rhythm.  It was found that frequent and sustained disruptions to the circadian rhythms can weaken the connection which weakens the shared signals. 

The mathematical model indicated that a small number of disruptions can make the connection closer but they are still working on how this relates to the fact that a lot of disruptions can cause a weakening of communications between the neurons.  Mathematical modeling allows scientist to study situations that cannot be explored through normal methods. It creates a situation where they can change variables without hurting anyone. 

This is interesting because I know from personal experience that jet-lag can really mess up your internal clock especially if you live in a location that does not have nights that get dark. I also find that my sleep patterns take a while to get back to normal.  Let me know what you think, I'd love to hear.  Have a great day.

Heading Home

 Heading home after month of travel. Back on Monday.

Roman Numerals

 

Today’s thoughts will be on the short side as I am getting ready to fly home from Sydney, Australia. Today, I explored a part of Sydney that was new to me. I found the courthouse in Darlinghurst, on Oxford street. 

As I looked at two of the buildings, I realized that they has]d the year of construction marked in Roman numerals. It took myself and my travel companion several minutes to determine the values. We are both pretty good up to 100 but a bit shakier with 500 and 1000.

Although we introduce students to Roman Numerals, we really don’t do much with then to place then in some sort of context. For instance, do we work with the history teacher when they are exposing students to Ancient Rome. This would provide them with a bit more context and students could see how they were used.

We could show a clip of a movie that shows the year of publication. I have not checked recently but several years ago you’d see the dates like MDCCCCLII. I don’t know if students have ever caught that. I noticed it and tried to figure out the year. In addition, some clock faces Wes done in Roman numerals but I am not sure how many now are done in this way.

The other place, I regularly see Roman numerals used is on buildings that have been around a while. As mentioned earlier, I saw them on a court building. 

As you can see 1884 and 1888 are the dates on these buildings. I have no idea why they used roman numerals but it may be either tradition, or they thought it made buildings look more official. Let me know what you think, I’d love to hear.

Growth Mindset In Math.

Last time, we looked at misconceptions regarding growth mindset versus fixed mindset. Today, we will look at the topic specifically in regard to math since it is a subject where many parents perpetuate a fixed mind set. We know that students with a fixed mindset believe that their abilities, intelligence, or talents are fixed and cannot be changed.  Students with a growth mindset believe this talents and abilities can be developed with good teaching, effort and perseverance.

It has been found that students who have a growth mindset are more willing to put in effort even when they struggle or fail. In addition, they are able to stay focused on what they are learning. Students with a growth mindset tend to do better in math and are more engaged in the lessons. 

There are ways to help students develop a growth mindset. First of all, Jo Boaler’s website Youcubed.org has some really nice material on working towards a growth mindset there. There are videos, printed materials and more that can be used in the classroom.  In addition, there are some great rich open ended tasks  on the site to be used in class.

In addition, there are things one can do to help encourage a growth mindset in our students. The first thing is to introduce students to the idea that the brain has the ability to grow. This can be done by showing a clip from the YouTube video “The Neuron Song” for younger students or a clip from the BBC documentary “The Human Body” for older students. This introduces students to brain neuroplasticity.

Second, show how mistakes can be used to promote brain growth. If you as the teacher make mistakes, take time to show students how you can learn from the mistake. Spend time during class to analyze typical mistakes and show them how to analyze their own mistakes. Create a culture that celebrates mistakes and learning from them.

Third, instead or relying on closed traditional problems, expose students to open ended, rich tasks. Ask students if they can solve a problem in two different ways. Learn how to take traditional problems and change them into open ended problems for students  to practice. Give them challenges such as take four 4’s and any operation to find 1 to 20. An example for that would be 4/4 times 4/4. That is not the only solution, just one.

Remove the emphasis on speed. Teach students strategies to use and emphasis that learning to use these strategies are more important than quickly finding an answer. Assign few problems and insist students justify their answers or find multiple answers. Throw in a couple of reflective questions that make them think about what they’ve learned, etc.

Finally, be mindful of your attitude and watch the wording you use in class. In addition, teachers need to make sure they do not show a fixed mindset even vaguely because students will pick up on that. If we want students to change their mindset, we have to model the appropriate ways so students see the growth mindset in action.

With the new school year starting, it is the perfect opportunity to begin the year with activities designed to help students develop a growth mindset. This is especially important because many students have gotten behind, are aware of it, and feel as if they cannot do the math. Let me know what you think, I’d love to hear. Have a great day.

Warmup.

If you get 250 peanuts per square foot, how many peanuts will you get from 43560 square feet?

Warmup

 

If you raise 6 peanut plants per square foot, how many plants will you need for one acre. (One square acre = 43560 square feet).

What Is Or Isn’t Growth Mindset.

 I was on a tour yesterday and ran into a young lady whose pre-algebra teacher did the growth mindset videos from Jo Boaler, I know that some people are confused about the term growth mindset as defined by Carol Dweck. In a nutshell, a growth mindset is one where the person believes that their talents can be developed through hard work, feedback, and the use of good strategies while some with a fixed mindset believe that only someone with innate gifts.

Unfortunately, the basic idea has been taken by the general public, twisted, adjusted, and changed it. So today, we’ll look at how people have begun interpreting growth mindset since the term has become such a buzz word in education and industry

Some people believe they have always had a growth mindset because they have always been flexible and / or have a positive outlook. This is referred to as a false growth mindset because they do not understand what a true growth mindset is. In addition, most people are a combination of fixed and growth mindset simply because a pure growth mindset does not exist. The mixture changes over time based on experiences.

Other people believe a growth mindset involves praising and rewarding effort but it does not work in school, nor does o]it work in industry.  Unfortunately, outcomes matter in both situations. If we are going to reward people, we need to reward them for effort, learning and process because unproductive effort is not a good thing. It does not produce a desired outcome. It is important for people to understand that the process which includes asking for help, trying something new, analyzing set backs to move forward effectively.

Espousing growth mindset is not enough to change anything. Most places have mission statements describing what they want to do in wonderful terminology but they have not implemented any policies to help folks attain those goals. They encourage risk taking knowing people might not reach the goals and they also reward people when they learn something even though the project goals were not met. They are willing to support collaboration throughout the organization instead of encouraging competition. They want every member of the organization to grow by providing opportunities.

These are the three area where misconceptions abound but even if all of these misconceptions are cleared away, it does not pave the way for growth mindset. This is because of a persons fixed mindset triggers such as received criticism, comparing ourselves poorly to others, or certain challenges. 

This is a general overall look at the topic. Monday, I will look at growth mindset and what can be done from the start of the year to encourage it.  Let me know what you think, I’d love to hear.

Covering As Much Material As Possible. - Pacing

 

We all know that every math book comes with a suggested pacing guide designed to get through just about every topic in the time available. The problem is that most students are not ready to move at that pace. If you have students who struggle, or have certain limitations, you just cannot teach at the pace suggested. I once had a principal who could not understand why my class ( filled with special ed students, students who were not up to grade level, etc) could not keep pace with the other class filled with gifted students. The other teacher had to explain why I couldn’t match her pace. No matter what, we want to teach students as much as possible, so today we are looking at ways to do that.

One of the first things to do is administer a protest to students. Most books have a pretest associated with the chapter. The results of the pretest give you an idea of which parts you can teach, which you can touch on, and which ones you can skip. The results also give you an idea of which students need scaffolding, or need to be challenged. 

As you teach the material, you might notice that some students did not fully get the material, look at the material in the rest of the book to see if there exists the opportunity to reinforce their learning by reteaching the topic in the future. If the opportunity does exist, then you can move on, revisiting the topic as needed.

Periodically sit down and see how far your pacing has taken you eithr]er individually or as a department. If you have not covered as much as you hoped, sit down and determine if there are concepts or topics that can be cut out. Sometimes, it is possible to combine material rather than teaching them individually. This should be done a couple times throughout the school year.

Another thing to look at is the structure of the lesson itself. Are you spending a third of the period going over questions, is the lecture taking up most of the class, or do they have enough time to work on the assignment. In this vein there are a few things one can do. It is important to know how much to]ime you actually have to do the lesson. Pacing with in the lesson is just as important as pacing through the year. When pacing is good, students are more likely to be engaged.

It is said that a well paced lesson leads to increased instructional time. In addition it is important to identify the specific learning target so you know what they need to know already and what they should have learned by the end of the lesson. In addition, plan the questions you want to ask ahead of time so the questions are specific to the learning goals. Also plan your teaching points so you stay on topic.

Finally, make sure the planned activities are ones that meet the time scheduled, help students learn the concepts. So pacing is both through the year and through the lessons. Let me know what you think, I’d love to hear. Have a great day.

More First Day Of School Activities.

 Today, we will be looking at more activities that can be done on the first day or two of school. I do not like starting immediately with the syllabus and regular class because students seem to take a day or two to transition back into tzhe swing of things. I addressed this last week and today, we will look at a few more activities. Although, these activities may not contain direct instruction, they will be mathematically based.

Maybe start with the odd one out game. In this game, you give students a card with numbers on it but one number is not part of the pattern. The student has to identify the odd one out and they have to explain what makes it the odd one out. An example might be, 1, 4, 9,16, 25, 35, 49. The odd one out is 35 because it is not a square number.

Another activity is to have students create three dimensional shapes from colored paper and tape. The shapes students can make might be cubes, prisms, cones, cylinders, or rectangular prisms. The students have to figure out how to create the shapes rather than giving them pre-done nets. Some students might get a bit frustrated but others will have a great time. This is an activity that can provide a situation for students to collaborate.

Another activity is designed to review fractions. The idea is to pass out different shapes and then ask students to color in a potion of it to illustrate a fraction. For instance, you pass out triangles and ask students to fill in 1/3rd of it. This is a good way to figure out if students understand that all parts need to be the same size. When you have students work with different shapes, they have to think about it. Most fractions are illustrated using circles or rectangles so when you give them a trapezoid, triangle, diamond, etc, they have to figure out how to subdivide it into equal parts. 

Pass out the dice and have students calculate probabilities. Start with one dice before moving to two dice. Of course, one can also calculate the probabilities for various cards in a card deck. On the other hand, pull out the fun data so students can practice bar graphs, pie charts, line graphs, etc. it is easy to find these types of activities online. I had my students do one on blueberry production in the United States and then world wide. 

In the past, I have passed out packets of M & M’s or skittles to the students. I had them sort them into colors , create a bar graph of all the colors before asking them to figure out the percentage each color is of the whole bag. I finished off by asking them to create a pie chart.  Another activity is to pass out a candy bar to each child and ask them to calculate its volume.  I am aware one has to be aware of allergies so it might not work at your school.

Finally, bingo. It might be a mean, median, or mode version, order of operations, fractions, decimals, percent conversion, or whatever topic you want. It is easy to do. Simply, pass out empty bingo cards, give a long list of possible answers, and let the students fill out their own cards, then it’s time for the equations or data so students find the answer. This activity has always generated lots of discussion 

So with this many suggestions, you have some great choices available to make the first day of school enjoyable. Let me know what you think, I’d love to hear, have a great week.

Warm-up

On average, a bee hive contains 50,000 bees. If they weigh a total of 12 pounds, how much does each bee weigh? 

Warmup

 If one bee collects 0.0288 ounce of honey, how many bees are needed to collect one pound of honey?

Geometry Of Bees.

 

Yes, we know that bees like to use the hexagonal shape due to its stability and strength but scientists have discovered another facet of bees and how they build their hives. In fact, both bees and wasps have managed to solve an architectural problem using geometry. The problem boils down to figuring out how to  increase the size of each cell and being able to combine the cells effectively. Nest building material is expensive and one cannot combine different sized hexagons into a single array.

So both bees and wasps came up with a unique solution. They produce both five and seven sided cells to fill in the blank areas created by using different sized hexagons. Being able to combine different sized hexagons is difficult but these creatures figured out a solution.  

Many of the colonies are run by female workers who raise the offspring of the queen in hexagonal shaped cells. At a certain point, the focus of the hive is to switch from raising offspring to raising new males and new queens. Since these bees are larger than workers, the hive needs larger cells. 

Scientists analyzed 115 images of various colonies by using an automated image analysis tool but the also verified all results by hand to double check everything. They counted the sides of the irregularly shaped cells that filled in the region between the smaller worker cells and the newer cells for male offspring and new queens.

The scientists discovered that the bees paired one 5 sided cell with a seven sided cell, repeating the pairing again and again to fill in the area. This pairing has the same number of open sides as pairings of hexagons. In fact the five side is made first and then the seven sided shape is created. Then a mathematician created a mathematical model for this and the results of the model indicate this is close to the optimal geometric solution. Thus, these creatures discovered s solution that is great. Let me know what you think, I’d love to hear. Have a great weekend.

Activities For The Beginning Of School In Math,

 

The summer is rapidly speeding by and the first day of school is not too far off. Often on the first day or two of class, I provide a few activities to get them back into school rather than diving straight in. I find it takes a couple days for students to transition into the right mind set.

One activity I enjoy having my students do is to think of something mathematical for each letter of the alphabet. I pass out a sheet of paper with instructions at the top and the alphabet down the side. They have say 10 minutes to come up with something math for each letter. For instance they might say add for A, or base for B. The time could be longer or shorter depending on your students. 

When the time is up, have students leave their answers on their desk and get up and look at other peoples answers. At the end of a minute (again, the time is up to you) they go back to their desks and fill out any missing letters. Then go through all the letters, having students share out answers that you write on the board. 

Want to get to know your students? Ask them to complete a more about me math activity. In this activity, students tell you all about themselves and their family but via mathematical equations. Instead of giving their age as 16, they might say my age is 4^2 or sqrt 64 x 2. They might tell you they have ( 5-4) brothers and 24/8 sisters. All numerical info must be conveyed using mathematical equations.

Set up a “which would you rather?” School based activity that has a bit of silly. A question might be “which would you rather”….. wear a superhero outfit to school or wear a cowboy hat with a fake mustache. It is not a serious game, just something to break the ice and add a twist of fun to the class.

Another fun activity is to create equation puzzles where each puzzle piece has equations or answers along the sides that match up. In groups students are expected to match up the equation with the correct answer without using a calculator or pen and paper. They are only allowed to talk about it. You can make puzzles out of a large triangle divided into smaller triangles or a large square or rectangle subdivided into smaller quadrilaterals. If you make them out of paper and laminate the pieces, you can use them every year.

Start the year off with writing prompts such as “I want to become better at math so I can ……” or “people who are good at math …..” or “ I used math this summer when I ……….”. Writing is an important way to communicate and it doesn’t hurt to begin having them do a night of writing on the first day.

Finally, think about emoji logic puzzles. The ones that have three lines of math but instead of numbers, there are emojis or other characters. An example might be:

😀 + 😀 + 😀 = 45

👀 + 👀 + 😀 = 35

👀 + 😀 + 🙈 = 30

Find the value for each character.

I will be back with some more activities you can use the first day of school. Let me know what you think, I’d love to hear. Have a great day.