Origami Pasta

 

Although this is not a straight math topic, it is quite fascinating and does involve behind the scenes math. I was listening to Science Friday and they had a segment on Origami Pasta which is a 3 D printed pasta.  It is printed flat but when placed in water, it changes into shapes.

Two scientists from Carnegie Mellon pondered the idea of having a pasta what could be purchased in a flat form and then changes to the more traditionally shaped pasta when cooked.  If they could do this, then pasta would take up to 60 percent less space in the shelves, and be easier to stack.

These researchers mapped out the various ridges on the flat pasta so that when it is placed in hot water, they've discovered that the pasta swells at different rates in the ridges and valleys.  Knowing how these ridges and valleys react, it is possible to have pasta curve into boxes, rose shaped flowers, and others.

This has not been the first look at this type of pasta.  The idea originated at MIT back in 2014 when certain scientists were working on a project and watched the Star Wars film in which Rey adds a powder to water and ends up with a fully baked loaf of bread.  They created a pasta film made out of a gelatin film and edible fibers.  The gelatin film is actually composed of two layers, one is the top dense layer with a bottom layer that is porous.

These researchers have been able to design it so the flat strands form into the shape of a flower, and pasta shapes such as rigatoni and macaroni.  In addition, they worked with a couple chefs in Boston to figure out ways of using the product in restaurants.  They ended up creating transparent disks of gelatin flavored with plankton and squid ink that immediately wrap around caviar.  They also made strips of noodles that separated when added to hot liquid.  Both tasted pretty good and the texture was great. 

The research was taken up by people at Carnegie Mellon materials research in which they would create among other things, a pasta that pops into its proper shape when added to water.  They are working with the pasta company Barilla who is providing Italian pasta flour so they can make a real pasta instead of something made out of gelatin. A pasta that tastes like it should, hold up like it should, and mix with sauces appropriately.

The process requires grooves to be placed in the pasta as it is made.  It is the positioning of the grooves which will determine how the pasta is shaped because the grooved area expands less than the smooth regions. If this is successful, it will revolutionize the pasta market because it will mean the pasta takes less space, so it can be transported and stored more efficiently.  

In addition, it has possible applications in other fields such as robotics and biomedical devices.  Let me know what you think, I'd love to hear.  Have a great day. 

Warm-up

Roll 4 dice and use the numbers facing up.  Then combine with 3 operations to find a total of 27.

Warm-up

Choose 4 different digits between 1 and 9 and use 3 operations to end with a total of 22.  You may use each digit only once, no repeating digits.

Levels Of Communication In Mathematical Discourse

 

As math teachers, we know we need to encourage mathematical discourse but as is often the case, we are told to do it but are not given the training to accompany the mandate.  I usually end up doing a bit more research to learn more about the topic so I can do a better job.  

Mathematical discourse is about helping students learn to talk about mathematics.  The discourse can involve six different forms of conversation but the choice of form indicates their level of mathematical literacy.  In the first level, they might use ordinary language which uses non mathematical language and vocabulary to convey ideas.  Instead of numerator, they might talk about the number on the top or division with the house. They use the language they have to communicate their ideas.

Next, they might use mathematical terms when they write or speak about ideas such as the cubed root of eight is two.  It is as if they can speak or write sentences that could be translated into symbols.  The third type of communication involves writing down the mathematical sentence using the proper symbols such as "x < 3" and are able to state that a number is less than three.They understand the connection between the symbols and the mathematical sentence.

The fourth type is the one where students are able to create models, diagrams, pictures or other visual representation to share mathematical ideas.  This connects abstract and concrete. Next is the ability to share unspoken assumptions or know what the non mathematical parts of a problem are such as when calculating the amount of sod needed for a 6 by 8 foot rectangular area and knowing what sod is.  Finally is the quasi-mathematical language are students who are missing certain words in their vocabulary, making it harder for them to express themselves mathematically.

Furthermore, the teacher can use these levels of communication to assess where the student is on the spectrum and to determine what they know or understand. It is the perfect opportunity to provide scaffolding to help students move from using non-mathematical language to using the proper terms and the associated representations.

As students develop mathematical literacy, their ability to engage in higher level discourse increases so they are better able to communicate their ideas, thoughts, and understanding.  For fluency students need to be able to create illustrations, drawings, or use manipulatives to provide a visual representation along with symbols, words, and correct vocabulary.

Next time, I'll look at ways to help students improve their literacy.  Let me know what you think, I'd love to hear.  Have a great day.

Personalizing Word Problems.

I know when I was in high school, I hated word problems.  Most of the problems I ran across were on situations, that were totally irrelevant to me.  I'd see problems that talked about going so many miles, across multiple states at a certain speed but I lived on an Island you could circle in less than one day. 

 Or those problems where one train left New York City at 8 am and another train leaves from Atlanta two hours later and trying to figure out when they would meet. I knew more about traveling by buses than I did in regard to trains. 

Then there were the names of the people in the word problems were John, Susie, or Linda when everyone I knew had other, more exotic names like Kiko, or Juan.  For most of my teaching career, I've lived in places with planes, boats, or snow machines and students with names like Dove, Apala, or Kanaya.  

I began rewriting word problems so my students could relate to them better.  I replaced the names with names of my students so they felt a more personal connection.  Instead of trains, which most students have never seen, let alone ridden, I used snow machines or ATV's so they knew more about the vehicle.  Most villages in Alaska are not connected to any other village by road so students don't always relate to diving 6 hours away by car but they might understand traveling 6 hours to go to another village to participate in a dance festival.

In addition, I have to change some of the prices listed because certain items always cost more in the villages.  For instance, gas tends to run $6 per gallon rather than $2 or $3 in other places.  You might spend $12 for four ears of corn or a pizza can run like double the normal cost.  

By rewriting the problems to use names and situations students are more familiar with, then are more likely to feel as if they are dealing with something they understand. They find it easier because they know the situations.  This way, they apply the mathematical concept to something they know, they understand, and can relate to fully.  Now there are some situations that are quite difficult to adjust for the bush and sometimes takes a bit of thinking but it is worth the effort because students are able to access prior knowledge and connect with the mathematical concept.

Once they've established that connection, it is easy to make minor changes such as having them use a car instead of a boat, change the speed of the vehicle, and work towards the original problem. Although we often work on word problems with students, it is usually just one of this and one of that which apply the mathematical concept but we seldom use multiple applications with variations for the same concept.  

I wish I could do more of these but time always seemed to be a limiting factor.  Let me know what you think, I'd love to hear.  Have a good day.

Increased Understanding Through Talking.

 

I really hate the way schools often look at ways to improve student scores on standardized test.  I've had to "teach to the test", analyze the questions to determine which strand the state test focused on and teach that, and even provide sample questions so students got used to testing type questions. Sometimes we look too much at the test that we forget other ways of helping students improve.

One such way is for teachers to create more situations where students work on talking to each other.  It might take the form of having students divided into pairs to work out different problems that have the same answer. If their answers agree, they did it right but if the answers are different, they can ask each other to explain how they did the work.  This often helps students find where they made a mistake.

Another way is to have students talk their way through word problems so they can identify and apply math terms rather than relying on a memorized process.  They need to discuss their math learning complete with concepts, use mathematical terms, while practicing verbal expression.  In addition, discussing the problems offers safe chances to undergo productive struggle and make mistakes.  

Furthermore, teachers need to include academic language while encouraging students to use it when discussing problems.  For instance, instead of talking about flipping the second fraction, we should be say we are multiplying by the reciprocal so students develop the language necessary to express themselves.

One suggested way of working through word problems, is to have students read the word problem in a small group of no more than four.  The students work on solving the problem together through discussion, expressing ideas, and trying the ideas to see if they work.  Once students have had a chance to work on the material, find an answer before the teacher takes time to explain how to do it.  The last step would be for students to share the way they got the answer.  This is important because it gives students a chance to share and supports the idea that their thoughts are important.  This type of approach can be applied to any topic such as having students work two step equations before they are "taught" how to do them.  

This type of approach does require a bit more planning than the traditional method.  When setting up the lesson plan, one has to identify the academic language students are expected to know at the end of the lesson, the content objective to accompany the language, and a social objective such as using Think-pair-share or group collaboration.  

Now I admit, it is hard to stand back and not give direct answers to student questions which tell them how to do it.  Many of us need to learn how to respond to student request for help so they can actually experience productive struggle.  My district requires we follow the textbook, it's pacing guide, and we don't give students enough time to actually experience something like this.  

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

Warm-up

This child is 13 months old, how old is he in hours?

Warm-up

This man is 68 years old?  How old is he in days?

Understanding Negative Numbers

 

I’ve noticed that many of my high school students have issues understanding that four minus five is the same as four plus a negative five.  To many students subtraction is a completely different problem from adding a negative number and they arrive in high school with that thought.

I wonder if it has to do with students in elementary school where subtraction is taught as a take away or removal such as in 43 - 12 = 31.  You started with 43 objects, took away 12 and are left with 31.  For this type of problem, students who are not fluent in  addition and subtraction facts will count on their fingers.

In addition, they are taught that we cannot subtract a larger number from a smaller one because they haven’t learned about negative numbers. This is because to take away a larger number of items from a smaller number means students have to learn about deficits and that can be a complex topic. 

So, once they are taught about negative numbers being on the left side of zero and positive numbers on the right, they have difficulty comprehending it.  Even when negative numbers are shown on a number line, students are not yet introduced to the idea that positive and negative signs indicate a direction.

In this situation a positive sign refers to one direction while the negative refers to the other direction.  If you look at a number line, positive has you moving right and a negative number has you going left.  This conflicts with the idea of taking away a certain number of items.

Then throw in the idea of subtracting a negative number turning into a positive value.  In this case, students often learn to change the double negative into a positive so they can add but they don’t know why.  I’ve heard it explained as the first minus sign says facing the negative direction, moving in the opposite direction or in the positive direction.  Other times, I’ve heard it explained as being the same thing as multiplying by a negative one.  

When you get up to more complex situations such as subtracting rational expressions.  In this situation, students treat the minus sign as a subtraction rather than the minus sign being applied to all terms that appear after it.

As far as the brain is concerned, much of the processing occurs in the Inferior Parietal Lobe for both positive and negative numbers.There appears to be increased activity in the area when the brain is working on subtraction problems.  In addition, schools do not seem to teach the use of negative numbers in the same way as with positive numbers.  They do not drill or have students practice problems using negative numbers in the same way they do with positive numbers.

It has been found the brain relies on procedural rules more often when working with a double negative such as 4 - (-4) and the brain often finds the bracketed negative sign confusing.  Furthermore, the brain tends to rewrite problems such as 7 + (-3) as 7-3 because it finds it easier to deal with.

Is there a way to help students become more fluent solving problems with negative numbers.  I don’t know but I do know the lack of fluidity slows students down and they find it harder to solve problems.  Let me know what you think, I’d love to hear.  Have a great day.

Math As A Language.

I've heard it said that we should teach math as if it were a foreign language.  For instance the language of math can have multiple meanings, be ambiguous, or have multiple interpretations for the same symbol.  Most of the time math is “spoken” only in an educational situation and is not a “first” language for people.  It has to be learned. Furthermore, there are the formal and informal versions. So what if instead of focusing on the concepts, we consider looking at math as an actual language with objects or nouns, and actions or verbs.  

You may wonder how verbs and nouns factor into mathematics since we usually break math content down into the concept and the process.  Think about it.  Nouns are the object or thing which means the concept is the same as the noun while the verb is action so the process is the verb as it is what we have to do.  The action might include communication, solving the problem, representations and models, interpreting the results.

By redefining the focus, we could turn the experience of learning into something students are more familiar with.  It would show students that learning mathematics is similar to learning another language rather than treating it as a separate entity.  

In math, most students have to go through two stages of decoding in order to determine what is going on.  First, they have to decode the spoken words to the proper context and then they have to translate how it is used within the context.  For instance, if you say the term whole numbers, the student might think of numbers with holes in them rather than positive integers and thus may come up with the wrong answer due to a misinterpretation of the original spoken word.

In addition, decoding in mathematics also includes symbols and signs which might be seen pictorially or verbally, might be associated with an operation, or even an expression.  Consequently, students need to learn the meaning of these  basic symbols and signs just as they learn those recommended sight words in elementary school.

As noted above, mathematical vocabulary can be confusing due to the differences between mathematical and non-mathematical definitions, the fact that two different symbols could refer to the same situation such as 3-5 is the same as 3 + (-5). Unfortunately, even graphical representations can cause confusion because of the difference in types and how they are interpreted.  For instance, a bar graph and a line graph represent different situations and are interpreted differently.  

It is important to help students learn the differences in mathematical context and by offering the possibility of teaching math by identifying verbs and nouns, they might find it easier to understand and adjust to the different contextual situations they experience in math.  Let me know what you think, I’d love to hear.  Have a great day.

Why Is Skip Counting So Important?

 

Since I teach high school, I don't understand why I get students who  use skip counting instead of multiplication.  In fact, I couldn't understand why it was used at all but after learning more about number sequencing, I am beginning to fathom why it is taught. Let's begin with the definition of skip counting itself.  Skip counting is when someone counts forward or backwards by units other than one.  It might be counting by two or five or any other number.  In reality, skip counting is just naming the multiples of a number, usually from least to greatest. When I was in elementary, we were told to count by 2s or 3s or what ever number.  It was never referred to as skip counting.

In math, skip counting is used to tell time, count money, and provides the foundation for other mathematical skills.   In addition, it provides strategies to use in addition, subtraction, multiplication or division. It also helps develop number sense so when students are ready move on to higher level mathematical concepts, they are able to make the transition. Skip counting can also be used with lowest common multiple, greatest common factor, factoring, prime and composite numbers, and fractions.

In reality, we use skip counting all the time outside of the classroom.  If my students have an event such as running the student store, they often arrange change in groups such as four quarters to make counting easier but they use skip counting to count the left over change.  When they are counting bills such as fives, tens, or twenties, they skip count as they look for the total. When something arrives such as a box of  books, we usually count by twos or fives to see if we got them all.  

Then in the classroom, one way we teach students to find the least common multiple is to list the multiples for each number until you see the same number in each list.  This is a practical application of skip counting.  Another way is to list multiple of numbers smaller than a certain number such as for 6 and 12 I can list the multiples of 2, 3, and 6 to see what the greatest common factor.  every time we list multiples in any situation like this, you are skip counting. When looking at a analog clock, we usually skip count in 5's until we get to the closest 5.  On the other hand, if we have a double line of students, we'll count by two's to make sure they are all there.  

Furthermore, we see skip counting in certain sports such as basketball.   Every time, the player makes a normal basket, two points is added to the score which is counting by two.  If the player makes a basket from the 3 point line, they get three points which is counting by three.  So yes, one can go back and forth but the point is that the person keeping score is skip counting from the current score and based on the type of basket, the skip counting begins from there.

Or if we are trying to figure out what the date is going to be in two weeks, we skip count by 7s.  If today is the 14th, then in two weeks it will be the 28th.  Yes, I've had to do that a few times, especially this year because the prom was May 7th, Graduation was one week later, May 14, and the kids last day of school will be the 21st of May.  Skip counting by 7.

So these are some reasons why learning to skip counting is so important.  It helps advance number sense while building a stronger mathematical foundation.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 What math can you find in the picture?

Warm-up

 What do you notice?  What do you wonder?

The Four Stages Of Number Sequences

I stumbled across this while trying to figure out why skip counting is considered a good way to help student thinking switch from additive to multiplicative.  I'll explain why I was researching that topic another time but in the article, they discussed the four stages of number sequences.  I'd never heard of it because it explains how students acquire certain numerical concepts in elementary school.

The first stage of number sequences is the initial number sequence.  If a student has not hit the initial number sequence, they are considered pre-numerical and who do not understand cardinality.  When a child reaches the initial number sequence, they are able to understand a number such as 5, explain their cardinality, and can count them in order as a result but not as an finial unit.  In other words if they counted the five objects in front of them and arrived at five, on could add four more objects and most children would just continue counting till they arrived at 9.  They often rely on touching the item as they count or use their fingers to keep track of the total number. 

The second stage is known as the tacitly nested number sequence (TNS) in which students develop the understanding that there are subsequences contained within larger sequences such as 1 to 5 is a subsequence within the sequence of 1 to 10.  They can count the numbers of the subsequence regardless of where it occurs within a larger sequence such as. counting beginning at 5 to 12 inside of the sequence of 1 to 20 and this is referred to as a composite unit.  

If a student learns to count using skip counting, they are learning to use a composite unit such as when they count 3, 6, 9..., they are using a composite unit of 3.  In this stage, students use the composite units while counting by may not be aware of which composite unit they will use prior to counting. In addition, they understand that 8 is 8 ones or 8 times a single unit at this point so they don't count by ones all the time.  

The third stage is the explicitly nested number sequence (ENS). In this stage students can recognize that six ones is the same as one six.  They also understand that four sevens is the same as four composite units and the composite unit is made up of seven objects.  So in this stage, they understand multiplicative situations.  Furthermore, they have an abstract understanding of composite units and that six items is the same as six ones.  They also see that five ones is the same as one five and vice versa, so they understand they are referable.  In addition they can figure out how many groups of four are in 24 but they might struggle with  how many equal groups of six can 24 be divided into.

The final stage is the generalized nested number sequence (GNS) are students who are fluent in the multiplicative situations.  They understand composite units and how smaller units can fit into larger ones such as with a composite unit of 3 used eight  times to make 24 or as eight composite units containing 3 items in each unit.  A child who is in the GNS stage will begin building exponential structures, and begin to understand other things such as Lowest Common Multiples and Greatest Common Factors.

Skip counting is one step along this journey of number sequences.  Let me know what you think, I'd love to hear.  Have a great day. 

Creating An Effective Mathematical Environment.

 

We all struggle with creating an environment that helps students step out of the "I can't" mindset to the "I can". Every year we spend time setting up the classroom with the idea that students will want to do math but there are things we can do to make it more inviting and more effective. We know that the lesson itself is important but we as teachers have a lot of impact on student learning.

One of the most important things is to establish relationships with students and parents beginning on the first day with icebreakers in the class and sending letters home to parents or taking time to call the parents. Spend the first day getting to know your students if they are new and renewing relationships with students you've had.  In addition, use humor, tech, or questions to get on their level and share classroom rules via memes or snapchat or instagram. Remember you can set up class twitter, instagram, or facebook pages where you can share class happenings with parents. Don't be afraid to start from the beginning because students may not know how to take effective notes or know how to read a textbook when they come to you.  

Allow students the opportunity to weigh in on certain decisions such as how they'd like to see the room arranged.  About half my class like being spread out along the walls facing the center of the room while the other likes rows so my room looks crazy.  If you can, give them a chance to select the problems they want to do.  For instance, if the page has 32 problems, let them choose the 15 they want to do. It makes them feel as if they have some control and they feel more successful.

For instance, we can take the time to accept student ideas by allowing  students a chance to explain their solution and how they arrived at it.  Asking how they came up with the solution is often times friendlier than asking a student to explain their thinking because they don't connect finding a solution with thinking about it.  In addition, by recognizing their solutions and giving them a chance to share their thinking, students begin to develop confidence in themselves and their thinking. In addition, students gain a better understanding of the concept or topic

It is important to pose interesting questions that grab student curiosity so they want to find an answer.  It might be a question concerning two type of paint, one that covers in one coat and costs more while the other requires two coats but costs less, or perhaps it deals with the best deal on a cup of coffee.  These type of questions require students to rely on their own mathematical abilities and prior knowledge to answer.  These questions show students that math is found outside of the classroom.

Furthermore, take time to show students that mathematics are connected to other disciplines such as architecture, engineering, business, sports, etc so they see that math in the real world is not necessarily as neat as it is in the problems from the text book.  You might also take time to find literature with mathematical topics to read as a way of showing students, math is found in stories too.  By showing students that math is found in other disciplines, they begin to see that math is all around us and not just in the classroom.  

One of the last thing is to make sure students see math expressed in more than one way.  For instance, when teaching binomial multiplication, one should include more than just the FOIL method because not all students get the hang of it.  I show my students the FOIL method, the box method, the distributive method, the vertical method, and a picture.  When I teach factoring of trinomials, I use the standard diamond method, the reverse method and the reverse box method along with a picture so they can choose the method that works best for them.  

So open up lines of communications, give students a choice, show them how math relates to the world, and use multiple methods to create an effective classroom environment in math. Let me know what you think, I'd love to hear.  Have a great day.

4 Best Practices in Math

I love looking at best practices because I always come up with new information. In addition, they are always changing and evolving as researchers learn more. Sometimes it takes a bit to implement new ideas especially if you work for a district who mandates teacher following the pacing guide or the curriculum and both do a disservice to the students.

One of the recommendations is for teachers to teach students at the grade level they are at even if they are missing a few skills.  If you put students in a lower mathematical level they might resent it especially if placement is based on test results.  There are ways to help students manage grade level material such as allowing the use of a calculator when combining like terms if they have difficulty with integers.  The calculator does the arithmetic while they work on learning the more advanced concept rather than getting stuck on the arithmetic they struggle with.

It is also important to show students how certain things relate across the curriculum.  For instance, in elementary math, students learn that fractions must have a common denominator for adding or subtracting and the same applies to rational express that are being added or subtracted.  In addition, multiplying binomials can be done in the same way as two digit times two digit numbers. Most textbooks do not take time to show the connectivity so students see each topic or concept as unique rather than connected.

Most textbooks come with assessments built in.  The series I use have a quiz scheduled about half way through the chapter and a test at the end. This is very traditional but it doesn't give the teacher much data and it is better to carry out mini assessments throughout the chapter.  Mini assessments give both the teacher and student a chance to check understanding.  

The assessment doesn't have to be formal like a quiz.  Instead it might be something like a thumb up or thumb down to show yes or no, or a showing of fingers indicating 1 finger says I don't know to 5 which says I could teach another student.These type of assessments give the student a chance to self assess because they have to determine their level of understanding in order to share it with the teacher.  

For something a bit more formal, exit tickets and journal entries provide a nice amount of information without being a "quiz".  Its also possible to give students a "quiz" made up of a couple of questions. I like giving five questions at the beginning of the week and letting students retake the quiz to work on only the questions they missed.  When they have done all five questions correctly, they are done with the quiz for the week.

It is suggested that students be allowed to work together on assignments and other activities.  Research shows that students who work together in collaborative groups tend to achieve more.  In addition, it appeals to the social nature of students and students are less likely to get stuck.  Any collaboration among students must be monitored otherwise one person does all the work while the other copies.  

On Wednesday, I'll address what is needed to create an effective mathematical environment in the classroom.  The classroom environment is important to helping students be successful.  Let me know what you think, I'd love to hear.  Have a great day.

Happy Mother's Day

 

Warm-up

 If 2.26 billion dollars were spent on Mother's Day flowers in 2007 and 2.66 billion dollars in 2021, what was the percent increase?

Ways To Improve Math Attitude.

I read articles on helping students change their mindset from closed to open because they learn better but sometimes it is difficult to do this, especially when they tell you that their parents couldn’t do math either, or they just aren’t smart enough.  I finally found some suggestions that can relate the theoretical things with reality.

For instance, we tell students that it is fine making mistakes.  When we make mistakes we learn but we don’t always share anecdotes with our students where we learned from our mistakes.  This is important because it shows students that their teacher is not the all-knowing math person rather they are just like them.  

It is important to share times when you miscalculated the wood you needed for a project and didn’t get enough.  Out here in the bush of Alaska that can create a problem because you can't just pop over to the store and pick it up because it has to be ordered in.  In addition to sharing the mistake, it is important to talk about how much it cost you beyond what it might have cost if you’d calculated it correctly in the first place and how you might have done it properly the first time.

Not all math teachers excelled at math when they attended school.  I didn’t.  It was only after teaching math for a few years that I became really good at it because I kept learning more every time I taught the material. I also take time to talk about how some students learn the material faster than others  but many times the ones who take longer to learn the material will remember it way longer. 

In addition, when students start getting frustrated with the math, suggest they take a short break to give them time to relax before trying it again.  Take time to ask the student what they’ve tried and ask them if there are other things they could try or what did they do so far?   Questioning helps students understand their choices better and often helps them see a way to complete the problem.

Finally, try to make each  lesson engaging so students want to do them.  Students get tired of the same old same old, here are the notes, here is the assignment, do it.  I like adding activities that still require students to solve  a bunch of problems but I put the problems with answers around the room but the answers listed are not the answers to the problem they are with.  They are from another problem, so students solve a problem, look for the answer.  If they find the answer, that means they have it right but if they don’t they go back and work on it till they get the correct answer.  My students love this game.

These are just a few ways to help students develop a positive attitude in math.  Let me know what you think, I’d love to hear.  Have a great day.

Ways to Improve Student Success In Math

 

As math teachers we are always looking for strategies to help students do better in math. Some of the time, students are missing foundational knowledge, other times, it is the mind set or it could be any one of a number of other reasons they struggle.  Fortunately, there are some strategies out there that students can use to help them do better.

First, give the students five minutes at the beginning of the class period to free write as a way of helping them settle down to get ready for the lesson.  Most students have a break between classes and often rush to make it to class on time.  The five minutes provides the time they need to get themselves together.

About half way through the class, students begin to show signs of minds wandering so they are not paying attention to the lesson.  A simple way to counter this is to have students answer a multiple choice question by going to a corner of the room that represents their choice.  This bit of movement helps them refocus while adding in a bit of movement.

Another way to help students is to help them build their confidence in regard to doing math.  Unfortunately, many students arrive in high school convinced they are missing the “Math gene” or just aren’t good at it. One way to counter this is to praise the effort they put into doing the problem.  It is important not to talk about how smart they are because that reinforces the idea of math can only be done by a certain type of student.  Take time to recognize their effort to reinforce the idea that one learns math by working hard.

Other ways to help students improve their math confidence is to listen carefully to them when they explain how they completed a problem.  One should use this as a way of assessing their understanding in an informal way but it also shows students you value what they have to say. The last thing is to use open ended questions which allow for more than one “correct” answer.

In addition, it is important to encourage students to ask for clarification if they don’t understand something.  It is possible to determine misunderstandings the student has when they explain how they completed a problem but they also need to learn that it is ok to ask questions when they don’t understand. 

Furthermore, one needs to emphasize conceptual understanding over procedure because one can do the procedure without understanding the concept being taught.  Sometimes teachers focus too much on how to solve a problem step by step rather than making sure they associate the concept with the process. 

One should also try to find problems that are authentic but peak student interest at the same time because these types of problems are more likely to engage them so they want to solve them.  Using real world problems tends to increase student understanding and interest.  Ask a question that requires them to decide which size is better for the money is more likely to get them interested instead of doing a straightforward cost per unit problem.  

Pacing Guides

 

After working for a district that insists on people following the textbook pacing guide for two years, I’ve decided that is a bunch of crud.  In all honesty, the pacing of the guide is such that students rush through the book without really learning that much.  In addition, there is no time to differentiate, scaffold, or help students who have learning disabilities.  Furthermore, most of the differentiation and scaffolding provided by the textbook comes in the form of the problems assigned at the end of the section.

The program comes with all the examples set to show students how to do the examples in the book step by step.  It allows students to work on assigned problems from the book online so all they do is put in the answers without a place to show work.  All the quizzes and tests are done and ready to give out.  Unfortunately, the digital presentation doesn’t always allow the teacher to show every single step in detail.  

I get frustrated because I have students who have gaps in their knowledge when they arrive in high school and I don’t have time to help them fill in the gaps.  The pacing guide is not set up to add in activities such as “search and rescue” where they get to move around the room while working through a set of problems. 

I don’t have time to try all the great ideas I read about because the district wants us to follow the guide.  The guide only allows a maximum of two days for any one topic which is not enough for students to really learn and the pacing is set to get through the whole book rather than meeting the needs of the student.  Yes, I’ve gotten in a bit of trouble for not “following” the pacing guide.

The middle school has an accelerated math class filled with students who can move faster while I have the students who have special needs, who have gaps in their fundamental knowledge, or are absent a lot.  I can’t move as fast as the other class yet I’ve heard about it.  I will admit, I try to choose a pace that works for my students so I’m always one to two chapters behind the other class but I don’t care.  

The reality is that most students cannot move at the pace laid out by the guide.  In fact, if we move too fast, students do not learn and retain the material so at a point in the future, they may get frustrated due to not being able to do the math.  I understand why the textbooks offer a pacing guide but I don’t understand districts that think the pacing guide is god rather than a suggestion. 

I prefer to go through the book at a pace that meets the needs of my students while allowing them to learn.  I prefer having time to work in activities to help retain and use the material at a future date.  I want to be able to move faster when I can but slow down when needed.  I want to put student learning first rather than the pacing guide because student learning is more important.  Let me know what you think, I’d love to hear.  Have a great day.

Warmup

 

If a cat averaged 6 kittens per litter, how many generations would it take for the population to reach a population of 150 cats?

Warmup

 

If a cat has an average litter of 4 kittens, how many generations will it take to get a population of 150 cats?