Thursday, February 4, 2021
Friday, January 29, 2021
Expect students to be able to model their mathematical thinking. This modeling can be exemplified through drawings or diagrams, charts or tables, equations, and representations. Here are some good questions/prompts to help students focus on modeling their thinking -
(1) Show me what you mean so I can see it.
(2) Which representation (objects, diagram, table, etc.) is most helpful to you? Why?
(3) Let's write an equation to represent this situation?
(4) What are the important quantities/numbers in this problem?
(5) Let's organize the data in this problem so we can examine it more easily.
The ability to model the math shows a good understanding of the concept, and also shows a deeper commitment to thinking. It is not always easy to model the math, especially at first, but keep at it. Students will benefit from seeing the teacher's model and will benefit from creating their own models.
Wednesday, January 13, 2021
The first standard in the Standards for Mathematical Practice is "Make Sense of Problems and Persevere in Solving Them." Here are some good questions and prompts to help students with this standard -
What do you think you have to find out?
Can you think of a similar problem you have solved?
What have you already tried?
Where are numbers in this problem? Are they important?
Where can you start working in this problem?
Retell this problem in your own words.
Can you use objects or pictures to help solve this problem?
Is your answer reasonable?
Students need to make sense of the work assigned to them in order to see the importance of the concept. Building perseverance is extremely important in mathematics (and everywhere!). Perhaps this list of questions and prompts will help students as they work on this standard.
Tuesday, January 12, 2021
Today's blog entry cites the work of Henningsen and Stein, "Implementing Standards-Based Mathematics Instruction."
One teacher behavior that can greatly impact learning is to ask questions ... specifically, asking students to explain their thinking. The lack of this behavior can also impact learning in a negative way.
"Teachers can promote sense-making and deeper levels of understanding by consistently asking students to explain how they are thinking about the task. Teachers may cut off opportunities for sense-making by hurrying students through the tasks, thereby not allowing them the time to grapple with perplexing ideas."
Teachers must help students connect models, symbols, and words. Teachers must continue to ask questions until deep understanding is evident. Always remember "wait time" and how important it is for students to have a bit of a struggle as they learn.
Teachers can support learning by:
Building lessons and tasks on prior knowledge.
Scaffolding student thinking.
Asking thought-provoking questions.
Sustaining pressure for explanation.
Modeling high-level thinking.
Do not stop asking questions! Encourage the students to ask questions, too.
Monday, January 11, 2021
Let's review -
This short phrase should be included in every math lesson. It is so important that it might need to be included twice!
First, start each lesson with a review of concepts previously learned. Those concepts can be from the previous lesson or lessons that occurred further back. Then, always review at the end of a lesson. Make sure to go over the important concepts again.
Review needs to be treated like formative assessment. Listen to what the students say as you review with them ... listen to their choice of words (math vocabulary) and how confident they appear with the content. Allow this to guide the next lesson.
Ask good questions during a review. Be specific and encourage lots of talking.
Sometimes it is a good idea to review DURING a lesson. Stop and gauge the students' responses to the instruction and consider engaging in a review activity right then. This might seem to slow down the pace of the lesson, but it is important to remember that we sometimes need to slow down in order to go fast - later!
We all know the importance of a good review, but we sometimes forget to do it every day. Get into the habit of frequently and consistently reviewing the content. That will be time well-spent.
Friday, December 11, 2020
Teachers sometimes worry that students will struggle with understanding fractions. The earliest experiences students have with fractions should be (simply - but not always simple) exposure to what a fraction is, as opposed to performing operations with fractions. Do not underestimate the importance of this exposure.
An idea to help with this early exposure is to examine "half" of something, and then to examine "half of a half." Using manipulatives, students can gain a firm understanding of "half" and "fourth." For example, folding paper in half, then in half again, is a good way to explain the concept.
When using a paper-folding technique, it is important to color or shade the half and the fourth. Do not assume students understand the concept otherwise. Saying to a child, "We folded the paper in half and this now represents two halves" might not make sense to every child. Coloring or shading one of the halves is necessary. Then, color or shade the fourth.
This is also effective when helping students understand that the larger the denominator, the smaller the fractional amount is. Many students naturally assume that one-fourth is MORE than one-half since four is greater than two.
Another good manipulative/representation for half of a half is the ruler or yardstick. The tiny marks on a ruler that represent halves and fourths are already there - clear for students to see.
When starting with fractions, start simple and make sure all models/representations are clear. Give it time. The later activities (for example, addition with fractions) will work out better if students gain these early understandings.
Wednesday, December 2, 2020
That first snowfall can be exciting for our students. Teachers have always been happy to see it, too.
The changing of the seasons and the beauty of nature is often discussed with our students. Watching the snowflakes fall is mesmerizing and it is not always easy to remain focused on academics when we are thinking about the snow piling up.
Math is everywhere ... so how can we incorporate math into this first snowfall? There are probably a few ideas circulating out there, from measuring the snow to recording data to talking about the geometry in a snowflake. Maybe snow can be used to talk about big numbers (altitude at a ski resort) or fractions (how many hours in a day did it snow?).
Ask students to create short story problems about snow and have them share their problems with each other. Finding solutions to student-created problems is a great way to encourage thinking and creativity, and it allows ownership for the students.
Even in our virtual classroom settings, we can use that first snowfall to generate excitement and fun.
*Note* This blog was not updated in November due to a technical glitch, but the author does plan on entering posts regularly now that the glitch has been resolved.