Monday, October 19, 2020

Introducing Variables

When students enter the middle-school grades and they see variables in math problems, it is sometimes intimidating.  Students do not, necessarily, quickly grasp what a variable means.

Elementary teachers can introduce variables to their students.  Even though the textbooks may not have variables as part of the problems, teachers can change it up a little and make a big impact.

For example, 4 + ____ = 9 can become 4 + n = 9.  Explain to the students that the "n" represents the missing addend, just as the blank does.  Finding "n" means finding the missing variable.

Another example is ____ + _____ = 8.  For this problem, the teacher could change it up a bit ... n + y = 8.  In that situation the variables are different letters, so the teacher could say, "The numbers represented by the variables must not be the same." [And, in that case, there is more than one correct solution.] n + n = 8 would mean the numbers are the same.  n + (n + 2) = 8 means add a number to a number that is two greater than the original number.

Not only do these activities introduce variables (letters) but they also allow good thinking and good discourse as the problems are being solved.  An important part of being a successful mathematician is the ability to think.  Do not overlook that aspect as lessons are prepared.

Friday, October 9, 2020

Some Advice for "Blended" Learning

 Many students are on a 'blended learning' schedule - meaning they have some days on-campus each week and some days off-campus each week.  This schedule will provide challenges for teachers and parents. Here are some words of advice as we face these challenges.  Perhaps these words will resonate.


(1) Help our children to develop an academic mind-set:  Encourage them to engage in pleasure reading (chapter books, newspapers, magazines), to practice their counting skills, to write in a journal each day (5-6 sentences will be enough), and to play games.  These activities help children to think and to focus.

(2) Encourage children to become aware of time and NOT to waste time:  While we are on a blended schedule, each minute counts.  Stay focused at school.  Don't procrastinate while at home.  

(3) Ask specific questions:  Do not ask a child, after completing an activity, "Did you understand that?"  Such a question only allows for a 'yes' or 'no' answer.  Rather, interact with children using more specific questions or requests, such as "Please explain to me how you solved that math problem ..." or "What is something you learned from this activity ...?"  These questions require children to become better at discourse and explaining, which are important skills.

(4) Take breaks: While working at home, take many breaks.  These should include a break from staring at the screen and should also include physical activities.  These breaks do not have to be brief, but even a brief break can be a good thing.  

(5) Realize the 'negative' emotions are natural:  Sometimes we will all feel frustrated, even defeated. Those are normal emotions during this challenging time ... normal for parents, teachers, and children.  We do not need to be ashamed of these feelings.  However, we must not stay in those negative places.  Try to keep a focus on the big picture - the learning.  When negative emotions enter the picture, start thinking of a new picture ... take a break ... do something pleasant and fun ... stretch and laugh and move away from the negativity.

Thursday, October 8, 2020

Time Spent and Time Spent Alone

Some children have difficulties in understanding elapsed time.  That is not an easy concept!

Consider having students write (on the top of their papers) the time in which they begin an activity and the time in which the activity ends.  This could become part of their regular routine and will open up lots of conversations about time spent on an activity.  The information can also be used to create story problems.

Children will begin to understand elapsed time better once they see those "time stamps" on their papers.  Additionally, they will begin to understand "wasted time," which is another important concept.

As students are often learning in isolation during these challenging days, it is a good opportunity to help them understand how time works.  A specific schedule set up for them will be helpful. For example, if they know a break occurs at 10:00 and lasts until 10:20, that allows opportunities to talk about "how long until ...?" or "how much more time do you need ...?" This might be more pertinent to children learning off-campus than to children who are in a traditional setting, as those learning off-campus must have some self-discipline and self-drive.

In addition to talking about minutes and hours, this pandemic allows good discussion to occur about days and weeks and months.  Ask students to create charts and tables and graphs that show data over the course of time.  

Time is another area in which number sense is vital.  When students struggle with telling time or elapsed time, check their abilities to count (by fives, for example) and use a clock as a guide.  

Wednesday, September 30, 2020

Small Intervention

Intervention is necessary.  Many students do not master a concept or miss some aspect of learning.  Teachers have to intervene, which might mean re-teaching or reviewing or taking another approach to the concept or learning target.  It can be a 'big thing.'

However, intervention can be effective if it is not "big" - "small" intervention can result in a big change for a student or a group of students.

Teachers often feel overwhelmed (which is understandable, considering all that is asked of them).  It is important to see intervention efforts as part of a daily routine, so that those efforts do not become overwhelming.  Here are a few tips for "small" intervention that will have a big impact -

(1) start and end each lesson with a brief review ... but ... make sure that review is specific and organized

(2) use "Number Talks" daily ... this can become formative assessment as well as intervention - a good "Number Talks" activity can be as brief as five minutes

(3) stop instruction and do a quick "check-in" with students - if there are questions or confusion, immediately address that with 'a different approach' - or - ask students to help each other by suggesting 'a different approach'

(4) divide the class into small groups and have them solve math problems on over-sized paper, utilizing two or three strategies ... then, as a group, they must choose one strategy and explain it to the class ... this allows discourse and encourages pride in work, but also allows for immediate intervention as the students will be forced to self-check and self-correct

Keep in mind that children do not always retain information as we hope they will, so each new day is an opportunity for review and re-teaching.  Keep those activities specific and clear. It's okay to think small when it comes to intervention.

Wednesday, September 16, 2020

Should I Purchase Flash Cards?

 Since many of our students are learning at home right now, some parents are wondering if old-fashioned flash card will help their children with addition facts and times tables.

Flash cards can have a place in a high-quality instructional environment.  They can be fun and there are some games that can be played with flash cards.

Keep in mind that "memorizing facts" is not enough for a child to have success in mathematics.  For example, a child might be able to recite a times table, let's say 3 x 4 = 12.  But if the child cannot relate those numbers to a real-life situation or make a connection to assist with problem-solving, then the fact has no purpose.  ((We have 3 students and each will be given 4 pencils. How many pencils will there be altogether?  What if the child says, "seven pencils" or what if the child says, "I don't know ... I don't know how to solve that problem"? In those cases, the child does not understand the math, so the memorization of a fact is not helpful.))

Some students are good at memorizing and that can make them seem capable of other required activities that are involved in math ... but that is not always the case.  Good mathematicians are capable of solving problems and they can create solution strategies that make sense and show high-level thinking.  Memorizing has its place and can be helpful, but it is not the ultimate goal.

If a parent does purchase a box of flash cards, he/she can use them with some level of effectiveness, but must be willing to move to higher-levels of thinking within the concepts.  Flash cards, in and of themselves, do not create a good instructional moment within mathematics.

It's What Counts

 The importance of counting is explained several times in this blog.  Once again, today's focus will be on that subject - encouraging and engaging in a daily counting activity.

With some students on campus and some students off-campus, this school-year will feel strange to everyone.  The instructional time we have with students is precious and important.  An 'extra' activity may feel unnecessary ... but do not think of a daily counting activity as unnecessary.  It is vitally important to help students develop number sense and confidence. 

Engage students in counting forward and backward -- by ones, twos, fives, and tens.  After a while, engage them in counting forward and backward by threes and fours and sixes and elevens. Do not always start at an 'obvious' spot -- for example, count forward by twos starting at seventeen, or count backward by tens, starting at 131.

A counting activity (like the ones described above and found elsewhere in this blog) should occur every day for students in grades k-8.  Even older students would benefit from such an activity.  Keep in mind, this activity should not take longer than a minute or two and does not have to be part of a Math Class setting ... it could be done at any time throughout the day.

Utilize counting games if the students feel bored with a traditional counting activity.  There are many great counting games and rhymes.

Students (especially in grades k-2) need to practice counting objects ... objects that are organized in a line or in a circle or in a random pattern.  This is another activity that should be done every day for maximized benefit.

Exposing our students to activities that build their number sense and help to increase their confidence with numbers is what counts!

Tuesday, May 12, 2020

Fraction Fun

Fractions are fun! They truly are!
(All math can be fun ... sometimes working with fractions might seem a little confusing, but encourage students to stick with it and they will gain the knowledge they need to be successful.)

Remember ...
Fractions show an amount.  Just like "three" is an amount, so is "three-fourths."
Fractions have a spot on the number line.
Fractions can be used to show distance or volume, just like whole numbers can.

One of the best ways to understand fractions and to use them effectively is with baking.  Following a recipe is a real-life example of how math "works" and many recipes involve fractions.  Doubling or halving a recipe is a great way to learn, as well.

A ruler is a good tool to help us understand fractions.  Study a ruler, then use the ruler to measure.  Writing the measurements helps us solidify our understanding of fractions.

Think about fractions and ratios.  Count the people in a room, then ask, "What number is wearing glasses and what number is not wearing glasses?"  Come up with a ratio and then make that a fraction.  Such an activity is a great way to learn how fractions can be used.

Do not fret about how operations work with fractions.  For example, some students get confused with adding fractions or multiplying fractions.  Focus on understanding what a fraction is before heading into the operations.

Do not let fractions frighten you!